1 00:00:00,060 --> 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 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,257 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,257 --> 00:00:17,882 at ocw.mit.edu. 8 00:00:21,312 --> 00:00:23,270 PROFESSOR: Why don't we go ahead and get going. 9 00:00:23,270 --> 00:00:25,420 Today what we want to do is talk about 10 00:00:25,420 --> 00:00:26,650 predator/prey interactions. 11 00:00:26,650 --> 00:00:29,393 In particular, how a predator and its associated prey 12 00:00:29,393 --> 00:00:32,860 can in principal oscillate over time. 13 00:00:32,860 --> 00:00:37,700 Now in this area there's this very important model block, 14 00:00:37,700 --> 00:00:40,311 a Volterra model, that has been used 15 00:00:40,311 --> 00:00:41,560 as kind of the standard model. 16 00:00:41,560 --> 00:00:43,851 It's given us some intuition for how this might happen. 17 00:00:43,851 --> 00:00:46,830 But mathematical biologists really 18 00:00:46,830 --> 00:00:50,165 don't like this model very much, because the oscillations 19 00:00:50,165 --> 00:00:54,010 that it give are neutral, or neutrally stable 20 00:00:54,010 --> 00:00:55,440 cycles or orbits. 21 00:00:55,440 --> 00:00:57,520 What I mean is that it's not a limit cycle. 22 00:00:57,520 --> 00:01:00,390 It doesn't have a characteristic amplitude, nor a period. 23 00:01:03,170 --> 00:01:05,420 And associated with that-- and it's not a concidence-- 24 00:01:05,420 --> 00:01:07,430 and associated with it, what it means 25 00:01:07,430 --> 00:01:10,060 is that if you make sort of very small modifications 26 00:01:10,060 --> 00:01:12,640 to the model, then you can get qualitatively different 27 00:01:12,640 --> 00:01:13,140 outcomes. 28 00:01:13,140 --> 00:01:16,470 In particular, you can either abolish the cycles-- i.e., 29 00:01:16,470 --> 00:01:19,130 you can turn these neutrally stable orbits 30 00:01:19,130 --> 00:01:21,190 into stable limits-- or I'm sorry. 31 00:01:21,190 --> 00:01:23,120 Into a stable spiral. 32 00:01:23,120 --> 00:01:25,824 So it may have damped the oscillations. 33 00:01:25,824 --> 00:01:27,990 So it could be that the oscillations kind of go away 34 00:01:27,990 --> 00:01:28,730 over time. 35 00:01:28,730 --> 00:01:31,680 Or it could be that you turn them into true limit cycle 36 00:01:31,680 --> 00:01:32,530 oscillations. 37 00:01:32,530 --> 00:01:35,290 And this could just be from very small changes in the model. 38 00:01:35,290 --> 00:01:37,900 That's associated with this fact that the logical Volterra 39 00:01:37,900 --> 00:01:40,217 model was neutrally stable. 40 00:01:40,217 --> 00:01:42,050 Then we'll kind of switch gears a little bit 41 00:01:42,050 --> 00:01:44,080 and talk about these experiments-- 42 00:01:44,080 --> 00:01:47,350 that Yoshida paper that you guys read-- where they looked 43 00:01:47,350 --> 00:01:50,310 at this question of what happens in kind of a laboratory 44 00:01:50,310 --> 00:01:53,450 experiment where you have a predator and its prey. 45 00:01:53,450 --> 00:01:54,920 And in particular what they found 46 00:01:54,920 --> 00:01:59,050 is that if there is evolution in the prey population, 47 00:01:59,050 --> 00:02:01,710 that you can get qualitatively different oscillations 48 00:02:01,710 --> 00:02:02,530 in particular. 49 00:02:02,530 --> 00:02:04,946 Instead of having a 90 degree phase shift between predator 50 00:02:04,946 --> 00:02:07,690 and prey, you can end up with 180 degrees 51 00:02:07,690 --> 00:02:11,830 phase shift, so kind of anti-correlated oscillations. 52 00:02:11,830 --> 00:02:14,400 But also that the oscillations could be much longer than what 53 00:02:14,400 --> 00:02:14,899 you'd. 54 00:02:14,899 --> 00:02:17,240 Expect 55 00:02:17,240 --> 00:02:18,840 And finally we'll say something about 56 00:02:18,840 --> 00:02:20,640 these noise-induced oscillations, which 57 00:02:20,640 --> 00:02:22,120 I believe that you guys have been 58 00:02:22,120 --> 00:02:28,540 playing with a bit in the context of your homework. 59 00:02:28,540 --> 00:02:29,670 Is it the next one? 60 00:02:29,670 --> 00:02:30,870 Oh, I get it. 61 00:02:30,870 --> 00:02:33,650 Well, you will get a chance to play with a great deal. 62 00:02:33,650 --> 00:02:35,240 So pay attention then. 63 00:02:35,240 --> 00:02:36,110 All right. 64 00:02:36,110 --> 00:02:39,290 Any questions before we get going? 65 00:02:39,290 --> 00:02:43,180 Problem set's due tomorrow so that you can enjoy Thanksgiving 66 00:02:43,180 --> 00:02:45,780 with your family or friends. 67 00:02:45,780 --> 00:02:47,506 And then we'll have one more problem set. 68 00:02:54,950 --> 00:02:56,480 So this Lotka-Volterra model. 69 00:03:01,350 --> 00:03:04,330 There's kind of a fun history of this, 70 00:03:04,330 --> 00:03:05,570 which I'll tell you about. 71 00:03:05,570 --> 00:03:09,857 But I just want to highlight that there 72 00:03:09,857 --> 00:03:12,190 are two important things that you should be remembering. 73 00:03:12,190 --> 00:03:16,680 And I'm getting these two points from Mark Kot in his book 74 00:03:16,680 --> 00:03:18,400 Elements of Theoretical Ecology, where 75 00:03:18,400 --> 00:03:21,150 he says there are two things you need to know about this model. 76 00:03:21,150 --> 00:03:25,290 One is that it's a bad model. 77 00:03:25,290 --> 00:03:27,890 And of course you can argue what you mean by bad. 78 00:03:27,890 --> 00:03:29,420 I think it's maybe overstating it. 79 00:03:29,420 --> 00:03:32,790 But we'll try to be explicit about why he might say that. 80 00:03:32,790 --> 00:03:35,500 But then it's sort of profoundly important. 81 00:03:38,790 --> 00:03:41,260 He kind of says from a historical perspective. 82 00:03:41,260 --> 00:03:44,840 But I would say that maybe both of these statements 83 00:03:44,840 --> 00:03:45,720 are sort of true. 84 00:03:45,720 --> 00:03:47,810 It's a bad model mathematically in some ways, 85 00:03:47,810 --> 00:03:50,570 because it's somehow structurally unstable, but also 86 00:03:50,570 --> 00:03:51,450 profoundly important. 87 00:03:54,150 --> 00:03:55,800 The mark of intelligence is being 88 00:03:55,800 --> 00:03:58,810 able to keep two incompatible ideas in your mind 89 00:03:58,810 --> 00:04:00,230 at the same time. 90 00:04:00,230 --> 00:04:02,120 Somebody said something like that. 91 00:04:02,120 --> 00:04:02,780 More than two? 92 00:04:02,780 --> 00:04:04,166 OK. 93 00:04:04,166 --> 00:04:06,290 Well, we'll try to figure out what we mean by this. 94 00:04:06,290 --> 00:04:07,750 All right. 95 00:04:07,750 --> 00:04:13,640 So Kot, it's a nice book on mathematical ecology, 96 00:04:13,640 --> 00:04:15,530 if you're curious. 97 00:04:15,530 --> 00:04:23,180 The history of this is that it was in the mid-1920s. 98 00:04:23,180 --> 00:04:26,860 And there was a marine biologist named Humberto D'Ancona. 99 00:04:26,860 --> 00:04:28,550 Does anybody speak Italian? 100 00:04:32,060 --> 00:04:33,550 Well, he was a marine biologist. 101 00:04:33,550 --> 00:04:37,190 And he had been studying the prevalence of different fish 102 00:04:37,190 --> 00:04:42,080 species in fish markets, and kind of throughout Italy. 103 00:04:42,080 --> 00:04:45,250 So he went to fish markets. 104 00:04:45,250 --> 00:04:52,600 Over the course of about R-K. For about 10 years. 105 00:04:52,600 --> 00:04:53,100 Right? 106 00:04:53,100 --> 00:05:00,300 So kind of roughly from 1912 to 1923. 107 00:05:00,300 --> 00:05:02,140 Something like that. 108 00:05:02,140 --> 00:05:04,010 And he basically asked, well, what 109 00:05:04,010 --> 00:05:05,720 is the kind of the composition of fish 110 00:05:05,720 --> 00:05:09,870 being sold in all these fish markets across Italy. 111 00:05:09,870 --> 00:05:12,700 And he noticed something that was very interesting, 112 00:05:12,700 --> 00:05:16,060 which is that there was a marked change in the composition 113 00:05:16,060 --> 00:05:19,910 between more predatory fish as compared 114 00:05:19,910 --> 00:05:21,650 to more prey-like fish. 115 00:05:21,650 --> 00:05:24,025 And this was associated with something that was happening 116 00:05:24,025 --> 00:05:25,360 in Europe around that time. 117 00:05:28,710 --> 00:05:30,130 World War I. OK. 118 00:05:30,130 --> 00:05:31,610 So World War I in the middle. 119 00:05:34,640 --> 00:05:39,120 And what he found was that in the middle of this period 120 00:05:39,120 --> 00:05:42,330 there was an increase, and then later a decrease 121 00:05:42,330 --> 00:05:50,601 of what he called selachians, which is apparently 122 00:05:50,601 --> 00:05:53,510 a word for sharks and shark-like fish. 123 00:05:57,354 --> 00:06:01,675 But for our purposes, we'll just say they are typically 124 00:06:01,675 --> 00:06:02,425 kind of predators. 125 00:06:06,020 --> 00:06:10,500 So what he saw is that this number of selachians, 126 00:06:10,500 --> 00:06:14,340 a function of time, kind of went like this, right. 127 00:06:14,340 --> 00:06:18,960 Where this was kind of roughly World War I. 128 00:06:18,960 --> 00:06:22,970 And he wanted to try to understand why did this happen. 129 00:06:22,970 --> 00:06:25,290 Why is it that war favors predators? 130 00:06:36,590 --> 00:06:39,700 Probably not just having to do with the general zeitgeist 131 00:06:39,700 --> 00:06:43,390 at the time.So it's probably a more mechanistic explanation. 132 00:06:43,390 --> 00:06:49,820 And luckily, Humberto was engaged to the beautiful Luisa 133 00:06:49,820 --> 00:06:50,320 Volterra. 134 00:06:50,320 --> 00:06:50,819 OK. 135 00:06:56,240 --> 00:06:56,780 Oops. 136 00:06:56,780 --> 00:07:03,240 Engaged to Luisa Volterra. 137 00:07:03,240 --> 00:07:09,490 So one day mid-1920s, our hero Humberto 138 00:07:09,490 --> 00:07:15,110 was visiting his girlfriend. 139 00:07:15,110 --> 00:07:19,720 And he started chatting with his future father-in-law, who 140 00:07:19,720 --> 00:07:24,240 was Vito Volterra, and asked him this question. 141 00:07:24,240 --> 00:07:28,110 And luckily, Vito Volterra was a famous mathematician. 142 00:07:28,110 --> 00:07:29,770 So he could write down a fancy model 143 00:07:29,770 --> 00:07:31,728 and try to get some insight into this question. 144 00:07:34,050 --> 00:07:37,620 So this is a case of random personal interaction leading 145 00:07:37,620 --> 00:07:38,840 to something interesting. 146 00:07:38,840 --> 00:07:42,950 That's the Volterra-- Lotka had actually 147 00:07:42,950 --> 00:07:47,360 studied these equations almost 15 years earlier. 148 00:07:47,360 --> 00:07:49,810 First in the context of auto catalytic kind of chemical 149 00:07:49,810 --> 00:07:50,900 reactions and so forth. 150 00:07:50,900 --> 00:07:53,800 But then later indeed, in the context of population dynamics. 151 00:07:53,800 --> 00:07:57,220 So Lotka wrote a book in the mid-1920s. 152 00:07:57,220 --> 00:08:03,297 Volterra wrote an article analyzing D'Ancona's data. 153 00:08:03,297 --> 00:08:05,380 And then now it's called the Lotka-Volterra model. 154 00:08:08,740 --> 00:08:10,300 And we'll come back to this thing 155 00:08:10,300 --> 00:08:13,770 about why is it that war favors predators after we 156 00:08:13,770 --> 00:08:15,580 get a sense of the model. 157 00:08:20,700 --> 00:08:24,650 So what Volterra wrote down was the following. 158 00:08:38,080 --> 00:08:40,590 First of all, we should all make sure we can figure out 159 00:08:40,590 --> 00:08:43,710 which one is the predator and which one is the prey. 160 00:08:43,710 --> 00:08:47,170 So let's think about this for 15 seconds. 161 00:08:47,170 --> 00:08:49,110 And then we're going to yell it out verbally. 162 00:08:58,400 --> 00:09:00,640 Is y the predator or the prey? 163 00:09:00,640 --> 00:09:02,899 Ready, three, two, one. 164 00:09:02,899 --> 00:09:03,690 AUDIENCE: Predator. 165 00:09:03,690 --> 00:09:04,523 PROFESSOR: Predator. 166 00:09:04,523 --> 00:09:05,220 All right. 167 00:09:05,220 --> 00:09:10,610 So we have a predator here and prey. 168 00:09:17,910 --> 00:09:19,960 Is the total size of the population 169 00:09:19,960 --> 00:09:23,790 constant? i.e, is x plus y equal to a constant? 170 00:09:23,790 --> 00:09:25,340 Ready, yes or no. 171 00:09:25,340 --> 00:09:25,840 Verbally. 172 00:09:25,840 --> 00:09:28,780 Three, two, one. 173 00:09:28,780 --> 00:09:31,260 No. 174 00:09:31,260 --> 00:09:35,520 It would be a constant perhaps if these two terms-- 175 00:09:35,520 --> 00:09:37,711 if b were equal to d or so. 176 00:09:37,711 --> 00:09:38,210 No. 177 00:09:38,210 --> 00:09:39,150 That's not even true. 178 00:09:39,150 --> 00:09:39,930 Never mind. 179 00:09:39,930 --> 00:09:40,430 Yeah. 180 00:09:40,430 --> 00:09:40,929 OK. 181 00:09:40,929 --> 00:09:43,810 I guess really what I'm thinking about is 182 00:09:43,810 --> 00:09:46,680 this tells us about how many predators can 183 00:09:46,680 --> 00:09:47,670 be created from a prey. 184 00:09:47,670 --> 00:09:49,700 But then there's also the growth in death rate. 185 00:09:49,700 --> 00:09:50,920 All right. 186 00:09:50,920 --> 00:09:55,010 But even this act of converting prey 187 00:09:55,010 --> 00:09:58,087 into a predator, even on that level 188 00:09:58,087 --> 00:09:59,170 it's not conserved, right? 189 00:09:59,170 --> 00:10:01,980 And of course, vegetarians like to point out 190 00:10:01,980 --> 00:10:09,420 that you can create many corn burgers for the price of making 191 00:10:09,420 --> 00:10:10,487 a hamburger. 192 00:10:10,487 --> 00:10:11,320 Something like that. 193 00:10:11,320 --> 00:10:11,820 OK. 194 00:10:14,420 --> 00:10:15,780 Except in the case of salmon. 195 00:10:15,780 --> 00:10:16,730 This is not true. 196 00:10:19,240 --> 00:10:20,380 Well, all right. 197 00:10:20,380 --> 00:10:23,000 We can analyze the ratio of d over b 198 00:10:23,000 --> 00:10:25,250 for a variety of different newt sandwiches 199 00:10:25,250 --> 00:10:26,410 after class, perhaps. 200 00:10:29,080 --> 00:10:29,780 OK. 201 00:10:29,780 --> 00:10:33,110 So this is pretty much the simplest model 202 00:10:33,110 --> 00:10:34,840 you can kind of possibly write down 203 00:10:34,840 --> 00:10:39,140 that captures this basic idea that when these two 204 00:10:39,140 --> 00:10:43,080 species interact, we're assuming that there's this mass action 205 00:10:43,080 --> 00:10:44,290 kind of rate. 206 00:10:44,290 --> 00:10:46,360 That it's just proportional to the number 207 00:10:46,360 --> 00:10:48,701 of the density of x and y. 208 00:10:48,701 --> 00:10:50,950 Now I think that in the context of chemical reactions, 209 00:10:50,950 --> 00:10:54,780 this is going to be true over a huge range of densities, 210 00:10:54,780 --> 00:10:57,700 and perhaps almost even rigorously true 211 00:10:57,700 --> 00:11:00,590 if they are just single. 212 00:11:00,590 --> 00:11:03,430 There's just some x and y that bounce into each other. 213 00:11:03,430 --> 00:11:06,140 And at some rate they do something. 214 00:11:06,140 --> 00:11:09,210 Whereas I think in the context of predator and prey, 215 00:11:09,210 --> 00:11:12,670 this is on much less firm footing. 216 00:11:12,670 --> 00:11:13,510 That's OK. 217 00:11:13,510 --> 00:11:14,870 So it's at least the simplest thing we can write down. 218 00:11:14,870 --> 00:11:16,810 And of course, then we'll have to ask, how is it 219 00:11:16,810 --> 00:11:18,185 that the conclusions of the model 220 00:11:18,185 --> 00:11:22,210 might vary depending on the assumptions that go in here. 221 00:11:22,210 --> 00:11:25,050 One thing that we've spent a lot of time in this class doing 222 00:11:25,050 --> 00:11:29,670 is trying to look at a set of equations, and in words, 223 00:11:29,670 --> 00:11:32,910 be able to extract what the assumptions were that went 224 00:11:32,910 --> 00:11:35,030 into writing the equation. 225 00:11:35,030 --> 00:11:36,690 And associated with that, you can 226 00:11:36,690 --> 00:11:39,660 think about all these things about the nondimensionalized 227 00:11:39,660 --> 00:11:41,080 versions of these equations. 228 00:11:41,080 --> 00:11:43,610 You'll often see the predator/prey models written 229 00:11:43,610 --> 00:11:48,980 in some nondimensionalized version, where for example, you 230 00:11:48,980 --> 00:11:52,230 get rid of maybe the a and maybe the c. 231 00:11:52,230 --> 00:11:53,760 I think that in many cases I prefer 232 00:11:53,760 --> 00:11:56,730 to just leave these terms in here, because that way we can 233 00:11:56,730 --> 00:11:59,639 immediately see what happens if we go hunting, 234 00:11:59,639 --> 00:12:00,680 or if we do this or that. 235 00:12:04,370 --> 00:12:05,110 OK. 236 00:12:05,110 --> 00:12:08,550 Can we try to figure out what-- there 237 00:12:08,550 --> 00:12:12,250 are four assumptions, at least, that 238 00:12:12,250 --> 00:12:16,180 have gone into this that are relevant and are worth saying. 239 00:12:16,180 --> 00:12:17,570 Can you guys help me out? 240 00:12:31,367 --> 00:12:31,867 Yeah. 241 00:12:31,867 --> 00:12:32,950 AUDIENCE: It's well-mixed. 242 00:12:32,950 --> 00:12:35,387 The interaction in terms of [INAUDIBLE]. 243 00:12:35,387 --> 00:12:35,970 PROFESSOR: OK. 244 00:12:35,970 --> 00:12:36,469 Right. 245 00:12:36,469 --> 00:12:37,490 So it's well-mixed. 246 00:12:37,490 --> 00:12:40,730 And there's two aspects that we might-- So one thing, 247 00:12:40,730 --> 00:12:44,310 it is certainly worth saying that it's well-mixed. 248 00:12:44,310 --> 00:12:46,360 But associated with that, you're going to say 249 00:12:46,360 --> 00:12:47,760 that it's because of x times y. 250 00:12:47,760 --> 00:12:50,120 Right? 251 00:12:50,120 --> 00:12:50,620 Yeah. 252 00:12:50,620 --> 00:12:53,470 I guess that's what I feel you were about to say. 253 00:12:54,919 --> 00:12:57,340 AUDIENCE: I mean, I guess. 254 00:12:57,340 --> 00:12:58,020 PROFESSOR: Yeah. 255 00:12:58,020 --> 00:12:58,460 Right I guess. 256 00:12:58,460 --> 00:13:00,668 But I think we have to keep these things a little bit 257 00:13:00,668 --> 00:13:05,730 separate in the context of chemistry, 258 00:13:05,730 --> 00:13:08,620 those are maybe the same statements, right? 259 00:13:08,620 --> 00:13:10,324 Of course, we're writing these things. 260 00:13:10,324 --> 00:13:12,740 These are just quantities that are just functions of time. 261 00:13:12,740 --> 00:13:14,670 We have not included space explicitly. 262 00:13:14,670 --> 00:13:17,110 So in that sense, it's definitely a well-mixed model. 263 00:13:17,110 --> 00:13:22,720 But just because we're assuming that it's well-mixed, 264 00:13:22,720 --> 00:13:24,834 we're not keeping track of the state of density 265 00:13:24,834 --> 00:13:26,250 as a function of position does not 266 00:13:26,250 --> 00:13:29,290 mean that they have to interact with this term. 267 00:13:29,290 --> 00:13:32,920 And for two molecules that are bouncing against each other, 268 00:13:32,920 --> 00:13:34,860 then I think that is the case. 269 00:13:34,860 --> 00:13:37,681 But we have to keep these assumptions separate. 270 00:13:37,681 --> 00:13:39,722 AUDIENCE: You could imagine a colony of bacteria, 271 00:13:39,722 --> 00:13:43,060 for example, where they're all-- you can only 272 00:13:43,060 --> 00:13:45,530 prey on the outside. 273 00:13:45,530 --> 00:13:46,650 PROFESSOR: Yeah. 274 00:13:46,650 --> 00:13:47,680 Necessary. 275 00:13:47,680 --> 00:13:51,702 I guess what I would say is that if it were not well-mixed, 276 00:13:51,702 --> 00:13:53,660 it could still be that they interact like this. 277 00:13:53,660 --> 00:13:56,200 And you just keep track of the predator/prey density 278 00:13:56,200 --> 00:13:57,620 as a function of position. 279 00:13:57,620 --> 00:14:00,270 And they could still interact in this way, possibly. 280 00:14:03,210 --> 00:14:08,370 And in particular, in the context of Turing patterns, 281 00:14:08,370 --> 00:14:10,880 what we did is we allowed these things 282 00:14:10,880 --> 00:14:12,574 to vary as a function of position. 283 00:14:12,574 --> 00:14:14,490 But then we would have still typically written 284 00:14:14,490 --> 00:14:17,330 the interaction term as x times y. 285 00:14:17,330 --> 00:14:21,186 So it is true we're modeling this 286 00:14:21,186 --> 00:14:22,310 as a well-mixed population. 287 00:14:22,310 --> 00:14:25,215 But I think that's independent of this statement. 288 00:14:29,030 --> 00:14:30,770 Do you agree or disagree? 289 00:14:30,770 --> 00:14:33,660 Or? 290 00:14:33,660 --> 00:14:35,100 AUDIENCE: But if it's well-mixed, 291 00:14:35,100 --> 00:14:41,340 the encounter frequency will be proportional to [INAUDIBLE]. 292 00:14:41,340 --> 00:14:43,194 AUDIENCE: That's sort of the secret, right? 293 00:14:43,194 --> 00:14:43,860 PROFESSOR: Yeah. 294 00:14:43,860 --> 00:14:46,644 AUDIENCE: The assumption is the encounter frequency 295 00:14:46,644 --> 00:14:49,124 is proportionate, right? 296 00:14:49,124 --> 00:14:49,790 PROFESSOR: Yeah. 297 00:14:49,790 --> 00:14:50,290 OK. 298 00:14:50,290 --> 00:14:55,020 So what I would say is-- well, certainly 299 00:14:55,020 --> 00:14:57,070 in the context of predators and prey, 300 00:14:57,070 --> 00:14:59,940 it could be that if the prey is full, 301 00:14:59,940 --> 00:15:04,650 then it's not going to be so simple. 302 00:15:04,650 --> 00:15:07,000 In a sense, is that I guess it could be more 303 00:15:07,000 --> 00:15:08,850 complicated than that is all. 304 00:15:08,850 --> 00:15:11,689 But my statement-- 305 00:15:11,689 --> 00:15:14,022 AUDIENCE: So you're saying-- I'm sorry-- the interaction 306 00:15:14,022 --> 00:15:18,330 term would not necessarily be just a function of x times y. 307 00:15:18,330 --> 00:15:19,500 PROFESSOR: That's right. 308 00:15:19,500 --> 00:15:21,370 It could be more complicated than that. 309 00:15:21,370 --> 00:15:24,030 And the other thing is that even if it were not well-mixed, 310 00:15:24,030 --> 00:15:27,127 in the context of these reaction diffusion models, 311 00:15:27,127 --> 00:15:29,710 we would typically say that the rate that these two things hit 312 00:15:29,710 --> 00:15:30,710 each other is x times y. 313 00:15:30,710 --> 00:15:32,360 But they're each a function of position. 314 00:15:32,360 --> 00:15:32,520 Right. 315 00:15:32,520 --> 00:15:34,680 So it's not that there's-- it can either be well-mixed 316 00:15:34,680 --> 00:15:35,130 or not. 317 00:15:35,130 --> 00:15:36,505 And it can either be this or not. 318 00:15:36,505 --> 00:15:38,167 I guess. 319 00:15:38,167 --> 00:15:40,125 AUDIENCE: But it's not quite the same variable, 320 00:15:40,125 --> 00:15:43,100 because in that case it would be the density, whereas-- 321 00:15:43,100 --> 00:15:44,220 PROFESSOR: I agree. 322 00:15:44,220 --> 00:15:46,850 Although here, it's not obvious whether we're 323 00:15:46,850 --> 00:15:50,220 talking about number or density, frankly. 324 00:15:50,220 --> 00:15:50,800 Right. 325 00:15:50,800 --> 00:15:55,040 And we often talk about this as if it's the number of predator, 326 00:15:55,040 --> 00:15:55,780 number of prey. 327 00:15:55,780 --> 00:15:58,350 But then that's a little bit inconsistent with this in that 328 00:15:58,350 --> 00:16:00,840 if you were-- this should really be, 329 00:16:00,840 --> 00:16:03,310 if you're thinking about these terms being the product. 330 00:16:03,310 --> 00:16:06,300 And it should really be the density. 331 00:16:06,300 --> 00:16:09,020 And if you're looking at the population 332 00:16:09,020 --> 00:16:13,547 size in some fixed area or volume, then it doesn't matter. 333 00:16:13,547 --> 00:16:15,380 But even in the context of predator or prey, 334 00:16:15,380 --> 00:16:18,160 I'd say that the way in which this would make senses 335 00:16:18,160 --> 00:16:18,976 is via densities. 336 00:16:24,910 --> 00:16:27,560 Does this discussion make sense? 337 00:16:27,560 --> 00:16:29,540 Maybe? 338 00:16:29,540 --> 00:16:32,320 We are assuming it's well-mixed, because we're not 339 00:16:32,320 --> 00:16:35,410 thinking about these things as a function of position. 340 00:16:35,410 --> 00:16:38,590 But I would say that we cannot get that from just 341 00:16:38,590 --> 00:16:41,256 the interaction term. 342 00:16:41,256 --> 00:16:42,922 We are assuming it's well-mixed, though. 343 00:16:47,059 --> 00:16:48,600 Other things that we've assumed here? 344 00:16:57,224 --> 00:16:59,182 AUDIENCE: Something along the lines of the prey 345 00:16:59,182 --> 00:17:01,110 doesn't need the predator, for instance. 346 00:17:01,110 --> 00:17:01,818 PROFESSOR: Right. 347 00:17:01,818 --> 00:17:05,119 The prey does not need the predator. 348 00:17:05,119 --> 00:17:06,910 And in particular, what happens to the prey 349 00:17:06,910 --> 00:17:08,140 in the absence of predator? 350 00:17:08,140 --> 00:17:09,819 AUDIENCE: It grows. 351 00:17:09,819 --> 00:17:11,660 PROFESSOR: So the prey grows exponentially. 352 00:17:11,660 --> 00:17:13,493 AUDIENCE: Because it's not diet [INAUDIBLE]. 353 00:17:19,880 --> 00:17:21,282 PROFESSOR: Without predator. 354 00:17:21,282 --> 00:17:22,990 And the other thing that is perhaps worth 355 00:17:22,990 --> 00:17:26,450 mentioning in all this though is that this 356 00:17:26,450 --> 00:17:29,880 is a deterministic description of the world. 357 00:17:29,880 --> 00:17:32,990 So a here is telling us about-- the normal way that we would 358 00:17:32,990 --> 00:17:37,570 interpret a is it's the rate of division of the cells, 359 00:17:37,570 --> 00:17:43,350 or the rate of new deer being born, or whatnot. 360 00:17:43,350 --> 00:17:44,830 But of course, a could actually be 361 00:17:44,830 --> 00:17:48,074 the difference between this growth rate and a death rate. 362 00:17:48,074 --> 00:17:50,240 So just because we don't have an explicit death rate 363 00:17:50,240 --> 00:17:52,410 doesn't mean that there's no death. 364 00:17:52,410 --> 00:17:59,369 So I think you can't actually say that the prey does not 365 00:17:59,369 --> 00:18:00,660 die in the absence of predator. 366 00:18:00,660 --> 00:18:04,920 What you can say is that the prey population-- 367 00:18:04,920 --> 00:18:06,700 we're assuming that a is greater than 0. 368 00:18:09,912 --> 00:18:12,120 So what you can say is that the prey population grows 369 00:18:12,120 --> 00:18:14,390 exponentially without the predator at rate a. 370 00:18:14,390 --> 00:18:16,590 But this is really in principle the difference 371 00:18:16,590 --> 00:18:19,940 between the growth rate and the death rate of the prey 372 00:18:19,940 --> 00:18:22,590 in the absence of the predator. 373 00:18:22,590 --> 00:18:24,710 And in the context of a differential equation, 374 00:18:24,710 --> 00:18:26,330 this doesn't make any difference. 375 00:18:26,330 --> 00:18:29,157 But if we were to go and do a master equation type formalism, 376 00:18:29,157 --> 00:18:30,365 would this make a difference? 377 00:18:32,961 --> 00:18:33,460 Yeah. 378 00:18:33,460 --> 00:18:33,960 Right. 379 00:18:33,960 --> 00:18:37,615 So this is the whole thing about the Fokker-Planck approach. 380 00:18:37,615 --> 00:18:41,880 And you can get different variances at equilibrium, 381 00:18:41,880 --> 00:18:44,880 depending on whether the rate of forward back reactions and so 382 00:18:44,880 --> 00:18:46,430 forth. 383 00:18:46,430 --> 00:18:48,983 Does that sound familiar to you guys? 384 00:18:48,983 --> 00:18:49,482 No. 385 00:18:54,874 --> 00:18:56,540 What else have we assumed in this model? 386 00:19:01,380 --> 00:19:03,316 AUDIENCE: The predator dies exponentially. 387 00:19:03,316 --> 00:19:04,024 PROFESSOR: Right. 388 00:19:04,024 --> 00:19:07,610 So the predator dies exponentially when? 389 00:19:07,610 --> 00:19:12,925 So the predator dies exponentially without the prey. 390 00:19:26,100 --> 00:19:27,485 So it's a nice simple assumption. 391 00:19:27,485 --> 00:19:29,860 Other things that you might like to point out about this? 392 00:19:32,668 --> 00:19:36,547 AUDIENCE: A single predator is [INAUDIBLE]. 393 00:19:36,547 --> 00:19:37,130 PROFESSOR: OK. 394 00:19:37,130 --> 00:19:37,629 Right. 395 00:19:37,629 --> 00:19:40,060 So there is a sense that it's just x times y. 396 00:19:40,060 --> 00:19:41,910 So the most simple way to think about this 397 00:19:41,910 --> 00:19:45,820 is that it somehow is a single predator eating a single prey. 398 00:19:45,820 --> 00:19:48,300 There's maybe no group hunting type behavior. 399 00:19:48,300 --> 00:19:52,530 So if we wanted to try to understand how a pack of wolves 400 00:19:52,530 --> 00:19:55,290 can bring down a buffalo, then you 401 00:19:55,290 --> 00:19:59,700 might not want to use this equation. 402 00:19:59,700 --> 00:20:03,870 So this is that the rate of predation 403 00:20:03,870 --> 00:20:12,930 is proportional-- we'll say goes as x times y. 404 00:20:12,930 --> 00:20:18,720 And that embodies many different kinds of assumptions. 405 00:20:29,371 --> 00:20:30,370 These are four findings. 406 00:20:41,860 --> 00:20:42,560 OK. 407 00:20:42,560 --> 00:20:45,440 So since it's such a simple model, we can go ahead 408 00:20:45,440 --> 00:20:46,630 and we can just solve it. 409 00:20:46,630 --> 00:20:47,130 Right. 410 00:20:49,670 --> 00:20:51,790 How many fixed points are there going to be? 411 00:21:06,120 --> 00:21:06,620 Two. 412 00:21:06,620 --> 00:21:08,100 OK. 413 00:21:08,100 --> 00:21:09,880 It's always good to remember these things. 414 00:21:09,880 --> 00:21:13,156 And so what we can think about, this as x. 415 00:21:13,156 --> 00:21:16,050 This is y. 416 00:21:16,050 --> 00:21:18,610 There's indeed going to be a fixed point somewhere in here. 417 00:21:18,610 --> 00:21:21,860 But then the other one is going to be at zero, zero. 418 00:21:21,860 --> 00:21:24,022 Because the top equation has x's everywhere. 419 00:21:24,022 --> 00:21:25,480 Bottom equation has y's everywhere. 420 00:21:25,480 --> 00:21:28,960 So indeed, there's going to be one fixed point at zero,zero. 421 00:21:32,620 --> 00:21:34,770 So-called the trivial fixed point. 422 00:21:37,610 --> 00:21:41,760 Is this a stable, or is this an unstable fixed point? 423 00:21:41,760 --> 00:21:43,260 Let's think about this for a moment. 424 00:21:43,260 --> 00:21:46,270 And then we'll vote verbally. 425 00:21:46,270 --> 00:21:48,160 Stable or unstable? 426 00:21:48,160 --> 00:21:48,660 OK. 427 00:21:48,660 --> 00:21:51,560 Ready, three, two, one. 428 00:21:51,560 --> 00:21:52,846 AUDIENCE: [INAUDIBLE] 429 00:21:52,846 --> 00:21:55,220 PROFESSOR: So I think there were some disagreements there 430 00:21:55,220 --> 00:21:57,620 probably. 431 00:21:57,620 --> 00:22:02,040 It was a little hard to tell from the verbal. 432 00:22:02,040 --> 00:22:04,670 And one way that we can think about this as that we 433 00:22:04,670 --> 00:22:08,150 can ask, well, the predator in the absence of the prey, what 434 00:22:08,150 --> 00:22:10,110 happens to it? 435 00:22:10,110 --> 00:22:14,290 So it kind of comes down here. 436 00:22:14,290 --> 00:22:17,550 So along this axis it's stable. 437 00:22:17,550 --> 00:22:20,060 However, in the absence of predator, 438 00:22:20,060 --> 00:22:22,460 if you think about the prey, that'll grow. 439 00:22:27,360 --> 00:22:29,620 So this is actually an unstable fixed point. 440 00:22:29,620 --> 00:22:33,471 Stable along one axis and unstable on the other. 441 00:22:33,471 --> 00:22:34,970 Incidentally, what does this tell us 442 00:22:34,970 --> 00:22:41,379 about the eigenvectors around this fixed point? 443 00:22:41,379 --> 00:22:42,990 AUDIENCE: [INAUDIBLE] 444 00:22:42,990 --> 00:22:43,990 PROFESSOR: That's right. 445 00:22:43,990 --> 00:22:47,275 The eigenvectors are really just 0, 1, 1, 0. 446 00:22:47,275 --> 00:22:49,650 Because you can see that if you start along one of these, 447 00:22:49,650 --> 00:22:50,360 you come straight down. 448 00:22:50,360 --> 00:22:52,443 Start along the other one, you come straight down. 449 00:22:56,320 --> 00:22:59,930 And the other one we can just see by-- well, 450 00:22:59,930 --> 00:23:02,160 we can solve it directly. 451 00:23:02,160 --> 00:23:03,630 Just we pull out an x. 452 00:23:03,630 --> 00:23:07,000 It's when a minus by is equal to zero. 453 00:23:07,000 --> 00:23:13,610 And also when minus c plus dx is equal to zero. 454 00:23:13,610 --> 00:23:18,790 So the other fixed point, this x star, y star, and this 455 00:23:18,790 --> 00:23:20,720 is maybe the important one. 456 00:23:20,720 --> 00:23:28,960 It's going to be just c over d and a over b. 457 00:23:36,130 --> 00:23:37,410 Super simple model. 458 00:23:37,410 --> 00:23:38,000 Right? 459 00:23:38,000 --> 00:23:40,340 You can calculate where the fixed 460 00:23:40,340 --> 00:23:42,630 points are kind of immediately. 461 00:23:42,630 --> 00:23:46,670 But it's actually a model that has many weird properties. 462 00:23:46,670 --> 00:23:50,640 Some of which you might think of as features. 463 00:23:50,640 --> 00:23:52,660 Some of which you might think of as bugs. 464 00:23:52,660 --> 00:23:58,800 But in particular, the location of this fixed point, I think, 465 00:23:58,800 --> 00:24:05,720 is really sort of surprising in that its dependence on this 466 00:24:05,720 --> 00:24:07,010 abcd. 467 00:24:07,010 --> 00:24:09,100 This is part of why I really think 468 00:24:09,100 --> 00:24:12,790 that when analyzing this model, I very much prefer 469 00:24:12,790 --> 00:24:15,450 to keep the a's, b's, c's, d's, rather than using 470 00:24:15,450 --> 00:24:17,200 the nondimensionalized version of it. 471 00:24:17,200 --> 00:24:21,160 Because when you do that, you lose track of what's going on. 472 00:24:21,160 --> 00:24:25,230 So this model for example, makes very clear predictions 473 00:24:25,230 --> 00:24:29,180 of what should happen if you hunt the predator. 474 00:24:38,910 --> 00:24:43,920 There are various contexts in which wildlife managers 475 00:24:43,920 --> 00:24:50,260 have been interested in trying to help a prey population. 476 00:24:50,260 --> 00:24:52,660 So if a prey population is suffering in some way, 477 00:24:52,660 --> 00:24:56,810 then you can reasonably think what you should want to do 478 00:24:56,810 --> 00:25:00,940 is kill the predator. 479 00:25:00,940 --> 00:25:04,850 And this is all these debates where you have the wildlife 480 00:25:04,850 --> 00:25:09,480 managers, where they buy automatic rifles in order 481 00:25:09,480 --> 00:25:12,010 to shoot wolves from helicopters. 482 00:25:12,010 --> 00:25:14,970 We have a Canadian in the room, right? 483 00:25:14,970 --> 00:25:17,454 I mean, wasn't this a Canadian proposal? 484 00:25:17,454 --> 00:25:18,620 AUDIENCE: I'm very American. 485 00:25:18,620 --> 00:25:19,785 PROFESSOR: All right. 486 00:25:19,785 --> 00:25:24,020 We'll have to do some more research to figure out exactly. 487 00:25:24,020 --> 00:25:24,520 OK. 488 00:25:24,520 --> 00:25:25,810 Right. 489 00:25:25,810 --> 00:25:28,680 So the question is, well, what happens at least in this model? 490 00:25:28,680 --> 00:25:31,020 And people apparently have seen this sort 491 00:25:31,020 --> 00:25:34,650 of thing occurring in natural populations as well, right? 492 00:25:34,650 --> 00:25:41,870 If you hunt the predator by going out and shooting them, 493 00:25:41,870 --> 00:25:44,580 what does this do to the predator and the prey 494 00:25:44,580 --> 00:25:46,440 populations. 495 00:25:46,440 --> 00:25:56,130 In particular, we think about x star, y star. 496 00:26:30,800 --> 00:26:33,345 Well maybe this-- am I confused? 497 00:26:40,950 --> 00:26:43,590 Maybe this is the example that I'm confused by. 498 00:26:43,590 --> 00:26:46,270 Well we have to complete it now that I started it. 499 00:26:46,270 --> 00:26:48,390 But now I might have gone backwards. 500 00:26:51,380 --> 00:26:53,660 We'll see what happens, and then discuss. 501 00:26:53,660 --> 00:26:54,160 All right. 502 00:26:54,160 --> 00:26:55,100 Ready. 503 00:26:55,100 --> 00:26:57,564 Three-- Oh wait. 504 00:26:57,564 --> 00:26:58,730 I don't have enough options. 505 00:26:58,730 --> 00:26:59,320 OK. 506 00:26:59,320 --> 00:26:59,820 Sorry. 507 00:26:59,820 --> 00:27:02,440 We should do one, then the other. 508 00:27:02,440 --> 00:27:04,690 Because sometimes things don't change, right? 509 00:27:04,690 --> 00:27:08,535 So how do I do this? 510 00:27:08,535 --> 00:27:09,320 OK. 511 00:27:09,320 --> 00:27:09,820 Sorry. 512 00:27:09,820 --> 00:27:10,320 OK. 513 00:27:10,320 --> 00:27:12,380 We should just do one, then the other, right? 514 00:27:20,300 --> 00:27:22,070 No change. 515 00:27:22,070 --> 00:27:23,810 This is down. 516 00:27:23,810 --> 00:27:24,310 OK. 517 00:27:27,995 --> 00:27:29,620 There's gonna be a problem [INAUDIBLE]. 518 00:27:36,490 --> 00:27:37,610 Let's first do y star. 519 00:27:37,610 --> 00:27:40,320 Cause that's the most direct one. 520 00:27:40,320 --> 00:27:40,820 Sorry. 521 00:27:40,820 --> 00:27:42,830 I think that my story I got backwards. 522 00:27:42,830 --> 00:27:43,914 But we'll figure this out. 523 00:27:43,914 --> 00:27:44,413 OK. 524 00:27:44,413 --> 00:27:44,930 Ready? 525 00:27:44,930 --> 00:27:45,430 OK. 526 00:27:45,430 --> 00:27:48,080 So the question is, there's a population 527 00:27:48,080 --> 00:27:49,300 that you want to-- all right. 528 00:27:49,300 --> 00:27:51,540 Now I'm going to redo my story to make it-- 529 00:27:51,540 --> 00:27:55,640 but there's a population you want to keep in check somehow. 530 00:27:55,640 --> 00:27:58,000 So you might reasonably go out and shoot it. 531 00:27:58,000 --> 00:27:59,355 Right? 532 00:27:59,355 --> 00:28:00,730 So the question is, if you go out 533 00:28:00,730 --> 00:28:03,400 and you do this on the predator, at least 534 00:28:03,400 --> 00:28:05,670 in this model, what happens? 535 00:28:05,670 --> 00:28:06,170 All right. 536 00:28:06,170 --> 00:28:06,740 Ready. 537 00:28:06,740 --> 00:28:08,940 Three, two, one. 538 00:28:08,940 --> 00:28:09,690 Right. 539 00:28:09,690 --> 00:28:12,739 So although we actually have a fair number 540 00:28:12,739 --> 00:28:13,780 of disagreements on this. 541 00:28:19,690 --> 00:28:21,810 So we're actually kind of 50-50 at this stage. 542 00:28:24,520 --> 00:28:28,280 What is it if you go out and from the helicopter, 543 00:28:28,280 --> 00:28:30,470 you start shooting this animal, what 544 00:28:30,470 --> 00:28:34,570 does that do from the standpoint of this model? 545 00:28:34,570 --> 00:28:35,070 Right. 546 00:28:35,070 --> 00:28:37,260 So we're shooting the predator. 547 00:28:40,170 --> 00:28:41,720 a, b, c, d, what you think? 548 00:28:41,720 --> 00:28:43,601 What should it change? 549 00:28:43,601 --> 00:28:44,100 c. 550 00:28:44,100 --> 00:28:46,330 And does c go up or go down? 551 00:28:46,330 --> 00:28:47,320 C goes up. 552 00:28:47,320 --> 00:28:48,010 OK. 553 00:28:48,010 --> 00:28:50,255 Well, how does that affect y star? 554 00:28:50,255 --> 00:28:51,130 AUDIENCE: [INAUDIBLE] 555 00:28:55,034 --> 00:28:56,010 PROFESSOR: Yeah. 556 00:28:56,010 --> 00:28:56,885 AUDIENCE: [INAUDIBLE] 557 00:29:00,694 --> 00:29:01,360 PROFESSOR: Yeah. 558 00:29:01,360 --> 00:29:01,860 Right. 559 00:29:01,860 --> 00:29:04,550 You know, I mean, I'm thinking daily helicopter rides 560 00:29:04,550 --> 00:29:08,185 where you shoot wolves as you see them. 561 00:29:08,185 --> 00:29:09,226 AUDIENCE: This is Canada. 562 00:29:09,226 --> 00:29:10,309 PROFESSOR: This is Canada. 563 00:29:10,309 --> 00:29:11,090 This is Canada. 564 00:29:11,090 --> 00:29:13,782 Yeah, those Canadians. 565 00:29:13,782 --> 00:29:16,973 AUDIENCE: So effectively, you're changing the death rate 566 00:29:16,973 --> 00:29:18,265 of-- that's what you're saying. 567 00:29:18,265 --> 00:29:19,098 PROFESSOR: Well, OK. 568 00:29:19,098 --> 00:29:19,900 So that's my claim. 569 00:29:19,900 --> 00:29:23,330 Of course, all of these things can be more complicated. 570 00:29:23,330 --> 00:29:27,470 But it certainly is change in the death rate 571 00:29:27,470 --> 00:29:30,580 when you go and shoot them. 572 00:29:30,580 --> 00:29:32,330 So I guess I would say that in the context 573 00:29:32,330 --> 00:29:34,705 of this model, that's the simplest way to think about it. 574 00:29:38,580 --> 00:29:42,397 So the statement is that if you hunt this predator, 575 00:29:42,397 --> 00:29:44,480 you're increasing the death rate for the predator. 576 00:29:44,480 --> 00:29:46,410 You're increasing c. 577 00:29:46,410 --> 00:29:47,950 Hunting predator. 578 00:29:47,950 --> 00:29:50,960 This causes c to go up. 579 00:29:50,960 --> 00:29:54,120 And the striking thing is that y star does not change. 580 00:29:54,120 --> 00:29:54,620 OK. 581 00:29:58,092 --> 00:30:00,076 AUDIENCE: I believe you. 582 00:30:00,076 --> 00:30:01,467 I think it's funny. 583 00:30:01,467 --> 00:30:02,050 PROFESSOR: OK. 584 00:30:02,050 --> 00:30:03,156 [LAUGHS] You believe me. 585 00:30:03,156 --> 00:30:04,279 OK, I'm glad. 586 00:30:04,279 --> 00:30:06,570 One thing is believing me in the context of this model. 587 00:30:06,570 --> 00:30:09,780 Another thing is asking whether this is happening in real life. 588 00:30:09,780 --> 00:30:12,460 But I think people have seen such weird phenomenon 589 00:30:12,460 --> 00:30:13,416 in the context. 590 00:30:13,416 --> 00:30:14,790 But then of course, it's a matter 591 00:30:14,790 --> 00:30:17,020 of is this a dominant source? 592 00:30:17,020 --> 00:30:17,740 What's going on? 593 00:30:17,740 --> 00:30:19,080 Like many, many things. 594 00:30:19,080 --> 00:30:19,881 Right. 595 00:30:19,881 --> 00:30:21,380 But certainly in this model, there's 596 00:30:21,380 --> 00:30:27,130 this weird phenomenon where the death rate of the predator 597 00:30:27,130 --> 00:30:32,360 does not alter this y star. 598 00:30:32,360 --> 00:30:32,860 OK. 599 00:30:32,860 --> 00:30:35,401 We have to figure out what the y star means here in a moment. 600 00:30:38,290 --> 00:30:44,360 What is the effect on this fixed point number 601 00:30:44,360 --> 00:30:47,810 concentration of the prey? 602 00:30:47,810 --> 00:30:49,030 It goes up. 603 00:30:49,030 --> 00:30:52,420 So my original story, I think I was getting confused. 604 00:30:52,420 --> 00:30:54,180 So at least in this model there's 605 00:30:54,180 --> 00:30:56,660 a sense that if you hunt the predator, 606 00:30:56,660 --> 00:31:00,210 it's true that you don't bring down the steady state, 607 00:31:00,210 --> 00:31:03,600 or the time average-- we'll see-- time averaged number 608 00:31:03,600 --> 00:31:05,300 of the predator population. 609 00:31:05,300 --> 00:31:08,190 But you do increase the prey population. 610 00:31:08,190 --> 00:31:10,277 Because in this case you can help the prey. 611 00:31:10,277 --> 00:31:11,860 But you don't bring down the predator. 612 00:31:15,960 --> 00:31:18,790 Now I want to say something more about justifying 613 00:31:18,790 --> 00:31:21,370 this thing about why we might care about x star and y star 614 00:31:21,370 --> 00:31:22,870 so much. 615 00:31:22,870 --> 00:31:24,460 So this thing is x star/y star. 616 00:31:27,374 --> 00:31:28,040 Well, maybe all. 617 00:31:34,771 --> 00:31:35,270 Yes. 618 00:31:35,270 --> 00:31:37,735 AUDIENCE: As soon as you stop shooting, 619 00:31:37,735 --> 00:31:40,693 this is a very temporary solution. 620 00:31:40,693 --> 00:31:43,170 When you stop shooting, c goes back to its original value. 621 00:31:43,170 --> 00:31:44,220 PROFESSOR: That's right. 622 00:31:44,220 --> 00:31:46,120 When you stop shooting, c goes back 623 00:31:46,120 --> 00:31:48,290 to its original model, or original value. 624 00:31:48,290 --> 00:31:51,020 And then the prey population will come back down. 625 00:31:51,020 --> 00:31:53,020 So this is something you have to continue to do. 626 00:32:00,720 --> 00:32:02,190 Yes. 627 00:32:02,190 --> 00:32:06,440 What do I wanna say? 628 00:32:06,440 --> 00:32:09,070 If you go ahead and you calculate 629 00:32:09,070 --> 00:32:14,170 the-- if we linearize around this fixed point x star/y 630 00:32:14,170 --> 00:32:17,240 star, that's our standard thing that we did early 631 00:32:17,240 --> 00:32:21,920 on the semester, what we find is that this Jacobian, 632 00:32:21,920 --> 00:32:27,000 this matrix A, the linearized matrix you get is-- all right. 633 00:32:27,000 --> 00:32:30,650 Well, we come here and we take the derivative with respect 634 00:32:30,650 --> 00:32:31,310 to x. 635 00:32:31,310 --> 00:32:32,485 So we get a minus by. 636 00:32:35,840 --> 00:32:39,000 Derivative of this guy up top with respect to y. 637 00:32:39,000 --> 00:32:40,440 And we get minus bx. 638 00:32:43,120 --> 00:32:47,900 And derivative of this g function with respect to x. 639 00:32:47,900 --> 00:32:50,520 We get dy. 640 00:32:50,520 --> 00:32:55,810 Now aspect to y, we have a minus c plus dx. 641 00:32:55,810 --> 00:32:59,870 Now we want to evaluate this around that fixed point 642 00:32:59,870 --> 00:33:07,410 at x star/y star, which is given here. 643 00:33:07,410 --> 00:33:09,920 We plug this in. 644 00:33:09,920 --> 00:33:11,220 So evaluate at y star. 645 00:33:11,220 --> 00:33:13,845 It's a over b 0. 646 00:33:17,340 --> 00:33:19,770 Evaluate at x star. 647 00:33:19,770 --> 00:33:22,280 This is just minus bc over d. 648 00:33:27,360 --> 00:33:29,960 Now y star, this is ab over b. 649 00:33:33,770 --> 00:33:35,410 And here we can get 0. 650 00:33:38,210 --> 00:33:42,510 So this is telling us about the linearized dynamics 651 00:33:42,510 --> 00:33:44,550 around a fixed point x star/y star. 652 00:33:48,590 --> 00:33:52,315 Now for this sort of matrix, what are the eigenvalues? 653 00:33:58,642 --> 00:34:01,660 It's not that the eigenvalues are 0. 654 00:34:01,660 --> 00:34:04,400 It's a slightly different statement. 655 00:34:44,679 --> 00:34:48,989 What are the eigenvalues for a matrix like this? 656 00:34:48,989 --> 00:34:49,620 What's that? 657 00:34:49,620 --> 00:34:50,800 AUDIENCE: They're imaginary. 658 00:34:50,800 --> 00:34:51,508 PROFESSOR: Right. 659 00:34:51,508 --> 00:34:53,610 They're purely imaginary. 660 00:34:53,610 --> 00:34:56,040 So the real part is equal to zero. 661 00:34:56,040 --> 00:34:58,309 So indeed, we can figure out this. 662 00:34:58,309 --> 00:35:00,600 Cause remember, to figure out what the eigenvalues are, 663 00:35:00,600 --> 00:35:04,500 you take the determinant of this vector 664 00:35:04,500 --> 00:35:08,015 a minus this lambda times the identity matrix. 665 00:35:10,600 --> 00:35:14,610 So we end up with, we want to take the determinant of-- we 666 00:35:14,610 --> 00:35:18,140 have a minus lambda, a minus lambda, 667 00:35:18,140 --> 00:35:19,530 and then minus bc over d. 668 00:35:29,367 --> 00:35:30,325 This is lambda squared. 669 00:35:33,970 --> 00:35:40,750 And then this is a plus, we have bc over d, ab over b. 670 00:35:49,300 --> 00:35:51,020 So we end up that the eigenvalues 671 00:35:51,020 --> 00:35:54,273 are equal to plus minus square root of a times ci. 672 00:35:57,570 --> 00:36:00,585 So they're purely imaginary eigenvalues. 673 00:36:07,711 --> 00:36:08,960 And what does this mean again? 674 00:36:25,800 --> 00:36:28,520 What can you say when you do this sort of analysis 675 00:36:28,520 --> 00:36:30,270 and you have purely imaginary eigenvalues? 676 00:36:37,136 --> 00:36:39,432 AUDIENCE: The orbits look like ellipses or something 677 00:36:39,432 --> 00:36:41,272 like that. 678 00:36:41,272 --> 00:36:41,980 PROFESSOR: Right. 679 00:36:41,980 --> 00:36:44,130 So the statement is that-- OK. 680 00:36:44,130 --> 00:36:47,310 And then we have to be careful in all of this business. 681 00:36:47,310 --> 00:36:49,210 So what we've done is we've taken 682 00:36:49,210 --> 00:36:51,770 a set of non-linear, a pair of non-linear differential 683 00:36:51,770 --> 00:36:52,340 equations. 684 00:36:52,340 --> 00:36:56,000 We've done the linear stability analysis. 685 00:36:56,000 --> 00:36:58,080 And we get purely imaginary. 686 00:36:58,080 --> 00:37:00,290 So one thing that you can say is that if you started 687 00:37:00,290 --> 00:37:02,770 with a linear system, and you got 688 00:37:02,770 --> 00:37:05,420 purely imaginary eigenvalues, that would tell you 689 00:37:05,420 --> 00:37:08,860 that indeed, you have these neutrally stable orbits, 690 00:37:08,860 --> 00:37:10,710 they go around. 691 00:37:10,710 --> 00:37:14,940 They can have a variety of different shapes. 692 00:37:14,940 --> 00:37:17,832 But given that the order that we did things in 693 00:37:17,832 --> 00:37:19,290 is that we took a nonlinear system, 694 00:37:19,290 --> 00:37:21,697 and we linearized, and then got this. 695 00:37:21,697 --> 00:37:23,030 What does that allow you to say? 696 00:37:32,730 --> 00:37:34,200 Does that prove that you actually 697 00:37:34,200 --> 00:37:39,410 have these neutrally stable [INAUDIBLE]? 698 00:37:39,410 --> 00:37:40,520 Yeah. 699 00:37:40,520 --> 00:37:43,120 Because this is one of those border cases, 700 00:37:43,120 --> 00:37:45,370 it unfortunately does not actually 701 00:37:45,370 --> 00:37:48,040 allow you to say that you have neutrally stable orbits. 702 00:37:48,040 --> 00:37:52,250 Because it turns out that the slight nonlinearities 703 00:37:52,250 --> 00:37:55,330 in the equations could cause problems, 704 00:37:55,330 --> 00:37:58,580 and cause it to go either to a stable limit cycle 705 00:37:58,580 --> 00:38:01,570 or to a stable spiral. 706 00:38:01,570 --> 00:38:03,790 The confusing thing is that in this case 707 00:38:03,790 --> 00:38:08,140 it is true that you have neutrally stable orbits. 708 00:38:08,140 --> 00:38:10,410 But that did not have to be true. 709 00:38:10,410 --> 00:38:15,010 And guys will get the chance to do this proof and so forth. 710 00:38:15,010 --> 00:38:17,625 Because there some conserved quantities. 711 00:38:17,625 --> 00:38:20,000 Of course, people have analyzed this thing in more depth. 712 00:38:20,000 --> 00:38:22,550 And it is true that you have neutrally stable orbits. 713 00:38:22,550 --> 00:38:24,820 But what is very hard to remember 714 00:38:24,820 --> 00:38:27,320 is that it didn't have to be the case just based 715 00:38:27,320 --> 00:38:29,910 on what we've said so far. 716 00:38:29,910 --> 00:38:30,410 Yeah. 717 00:38:30,410 --> 00:38:32,535 AUDIENCE: So you mentioned, it's not a coincidence. 718 00:38:32,535 --> 00:38:35,180 There is a deeper reason why? 719 00:38:35,180 --> 00:38:36,522 PROFESSOR: Well, coincidence. 720 00:38:36,522 --> 00:38:37,980 I guess what I'm saying is that you 721 00:38:37,980 --> 00:38:40,700 can prove that this thing does have 722 00:38:40,700 --> 00:38:43,860 these neutrally stable orbits. 723 00:38:43,860 --> 00:38:46,090 But we have not proven that. 724 00:38:46,090 --> 00:38:47,382 And we're not going to. 725 00:38:50,590 --> 00:38:55,180 Do these orbits go around clockwise or counterclockwise? 726 00:38:55,180 --> 00:38:55,680 Ready. 727 00:38:55,680 --> 00:39:00,475 Three, two, one. 728 00:39:00,475 --> 00:39:01,600 AUDIENCE: Counterclockwise. 729 00:39:01,600 --> 00:39:02,700 PROFESSOR: Counterclockwise. 730 00:39:02,700 --> 00:39:03,199 Right. 731 00:39:03,199 --> 00:39:06,554 And you can get that from thinking about what 732 00:39:06,554 --> 00:39:07,720 predator and prey should do. 733 00:39:07,720 --> 00:39:09,916 But also I've already drawn some arrows here. 734 00:39:09,916 --> 00:39:11,790 And that kind of helps provide some guidance. 735 00:39:15,370 --> 00:39:17,980 So they kind of come around like this. 736 00:39:17,980 --> 00:39:23,360 Now is this the only orbit that I should be drawing here? 737 00:39:23,360 --> 00:39:24,280 No. 738 00:39:24,280 --> 00:39:24,990 Right. 739 00:39:24,990 --> 00:39:28,120 Because it turns out that in this case 740 00:39:28,120 --> 00:39:30,670 there are an infinitely large number 741 00:39:30,670 --> 00:39:34,238 of orbits that could possibly come around. 742 00:39:39,451 --> 00:39:39,950 Yes. 743 00:39:39,950 --> 00:39:40,825 AUDIENCE: [INAUDIBLE] 744 00:39:44,340 --> 00:39:45,020 PROFESSOR: Yeah. 745 00:39:45,020 --> 00:39:48,157 Well, in the context of our simulation, 746 00:39:48,157 --> 00:39:50,240 the simplest thing that you guys would probably do 747 00:39:50,240 --> 00:39:52,300 is you would simulate and see what comes around. 748 00:39:52,300 --> 00:39:54,290 But yeah. 749 00:39:54,290 --> 00:39:56,767 I think even in the reading, did they prove it? 750 00:39:56,767 --> 00:39:59,350 Different authors in different books either prove it or don't. 751 00:39:59,350 --> 00:40:02,040 And there are a number of different-- 752 00:40:02,040 --> 00:40:03,930 you'll see in the problems, you can 753 00:40:03,930 --> 00:40:08,440 find that there is a quantity that is conserved 754 00:40:08,440 --> 00:40:09,870 along the equations of motion. 755 00:40:15,020 --> 00:40:17,821 But the quantity's different for each of these orbits. 756 00:40:17,821 --> 00:40:18,320 Right. 757 00:40:18,320 --> 00:40:19,820 So it's really saying that as you go, 758 00:40:19,820 --> 00:40:21,236 there's some quantities conserved. 759 00:40:21,236 --> 00:40:23,980 And it still has the same value when you go around. 760 00:40:23,980 --> 00:40:25,896 And so that means you had to kind of come back 761 00:40:25,896 --> 00:40:28,040 to where you were. 762 00:40:28,040 --> 00:40:31,077 So there is a conserved quantity. 763 00:40:31,077 --> 00:40:31,660 You'll see it. 764 00:40:37,680 --> 00:40:40,230 Now these neutrally stable orbits 765 00:40:40,230 --> 00:40:42,170 are kind of funny in several ways. 766 00:40:54,180 --> 00:40:57,470 Well, first of all, this eigenvalue 767 00:40:57,470 --> 00:41:00,740 tells you about the period of the orbits 768 00:41:00,740 --> 00:41:02,420 when you're close to the fixed point. 769 00:41:05,917 --> 00:41:07,500 And you can see that it doesn't depend 770 00:41:07,500 --> 00:41:08,625 on everything in the model. 771 00:41:08,625 --> 00:41:11,530 It depends on a and c. 772 00:41:11,530 --> 00:41:13,110 Does that kind of make sense, maybe? 773 00:41:17,420 --> 00:41:17,920 Yeah. 774 00:41:17,920 --> 00:41:18,580 Sort of. 775 00:41:18,580 --> 00:41:22,270 Cause a is telling us about how rapidly we grow up here. 776 00:41:22,270 --> 00:41:25,190 c is talking about how rapidly we die over here. 777 00:41:25,190 --> 00:41:27,930 So you know, it has something that at least 778 00:41:27,930 --> 00:41:30,850 has units of 1 over time. 779 00:41:34,120 --> 00:41:36,560 It makes sense that a and c should appear here. 780 00:41:36,560 --> 00:41:40,669 But I would submit that it's not totally obvious that b 781 00:41:40,669 --> 00:41:41,960 and d should not appear at all. 782 00:41:47,960 --> 00:41:51,230 So neutrally stable orbits, there's 783 00:41:51,230 --> 00:42:04,000 no characteristic amplitude to the oscillations, nor period. 784 00:42:07,410 --> 00:42:11,580 And this is at least true far from the fixed point. 785 00:42:11,580 --> 00:42:13,986 This tells you about maybe these orbits. 786 00:42:13,986 --> 00:42:16,360 But then it doesn't tell you about what happens far away. 787 00:42:20,640 --> 00:42:23,110 And these are not the kind of oscillations 788 00:42:23,110 --> 00:42:25,230 that from a mathematical standpoint 789 00:42:25,230 --> 00:42:28,000 we like the most, which are limit cycles. 790 00:42:28,000 --> 00:42:33,810 And remember, a limit cycle would look like this. 791 00:42:33,810 --> 00:42:35,450 If it were a limit cycle, it would 792 00:42:35,450 --> 00:42:39,390 be that there's some orbit that is stable in the sense 793 00:42:39,390 --> 00:42:41,810 that if you start inside of it, then 794 00:42:41,810 --> 00:42:43,450 you would approach it over time. 795 00:42:43,450 --> 00:42:45,117 If you start outside of it, then again, 796 00:42:45,117 --> 00:42:46,450 you would approach it over time. 797 00:42:50,070 --> 00:42:54,670 And this orbit then would have a characteristic amplitude, 798 00:42:54,670 --> 00:42:59,909 and a characteristic period, because it's one orbit. 799 00:42:59,909 --> 00:43:01,950 Whereas here, this has kind of an infinite number 800 00:43:01,950 --> 00:43:03,039 of different orbits. 801 00:43:03,039 --> 00:43:04,580 What this means is that just any sort 802 00:43:04,580 --> 00:43:07,180 of noise-- demographic fluctuations and so forth-- 803 00:43:07,180 --> 00:43:10,010 would cause the system to drift over time in terms 804 00:43:10,010 --> 00:43:13,044 of this amplitude of period. 805 00:43:13,044 --> 00:43:14,919 So it's true that if you got it started 806 00:43:14,919 --> 00:43:16,460 in the absence of any noise, it would 807 00:43:16,460 --> 00:43:19,750 keep on doing that oscillation forever with that amplitude. 808 00:43:19,750 --> 00:43:23,220 But any random noise will cause the thing to drift. 809 00:43:36,470 --> 00:43:38,295 The other thing that is worth stressing 810 00:43:38,295 --> 00:43:42,110 in this is that if you do a time average of the predator 811 00:43:42,110 --> 00:43:45,890 concentration, or the prey concentration, 812 00:43:45,890 --> 00:43:57,820 so if you do the average or the mean, x as a function of time, 813 00:43:57,820 --> 00:44:03,910 or y is a function of time is indeed x star y star. 814 00:44:03,910 --> 00:44:07,180 That's also something that you'd have to show. 815 00:44:07,180 --> 00:44:09,120 Not at all obvious from this. 816 00:44:09,120 --> 00:44:11,840 It's not a surprise for these orbits that are out here that 817 00:44:11,840 --> 00:44:13,620 the time average is indeed x star /y star. 818 00:44:13,620 --> 00:44:17,380 But for these big orbits, they probably already 819 00:44:17,380 --> 00:44:19,050 didn't have to be. 820 00:44:19,050 --> 00:44:21,666 So this is telling us that the analysis that we were talking 821 00:44:21,666 --> 00:44:23,040 about-- well, what happens if you 822 00:44:23,040 --> 00:44:26,990 do one thing or another thing-- that's actually 823 00:44:26,990 --> 00:44:30,360 kind of a reasonable quantity to be trying to calculate, 824 00:44:30,360 --> 00:44:34,760 because that is the average or the mean number 825 00:44:34,760 --> 00:44:36,140 of predator/prey in this model. 826 00:44:38,740 --> 00:44:39,560 Yes. 827 00:44:39,560 --> 00:44:42,440 AUDIENCE: So in a zombie apocalypse, 828 00:44:42,440 --> 00:44:45,330 how do we thin down the zombie [INAUDIBLE]. 829 00:44:45,330 --> 00:44:46,060 PROFESSOR: Yes. 830 00:44:46,060 --> 00:44:47,550 That's a very important question. 831 00:44:47,550 --> 00:44:50,610 In the event of a zombie apocalypse, 832 00:44:50,610 --> 00:44:53,400 what do we have to do in order to thin out 833 00:44:53,400 --> 00:44:54,810 the zombie population? 834 00:44:54,810 --> 00:44:55,310 All right. 835 00:44:55,310 --> 00:44:58,676 Well, the zombie movies typically 836 00:44:58,676 --> 00:44:59,800 are not yet at equilibrium. 837 00:45:02,720 --> 00:45:07,520 But certainly at equilibrium, what do you want to do? 838 00:45:10,040 --> 00:45:14,160 This is not a formal recommendation, by the way. 839 00:45:14,160 --> 00:45:16,770 But in this model, if you want to decrease 840 00:45:16,770 --> 00:45:21,576 the number of the predator, what is it that we want to do? 841 00:45:27,810 --> 00:45:28,310 Right. 842 00:45:28,310 --> 00:45:28,840 Well, OK. 843 00:45:28,840 --> 00:45:31,690 So in this model, if you want to decrease 844 00:45:31,690 --> 00:45:34,000 the number of the zombies at equilibrium, 845 00:45:34,000 --> 00:45:41,124 you want to decrease a, which corresponds to-- 846 00:45:41,124 --> 00:45:42,576 AUDIENCE: So it's a is equal to 0. 847 00:45:42,576 --> 00:45:44,512 So the birth rate is equal to the death rate, 848 00:45:44,512 --> 00:45:46,944 then there aren't new zombies. 849 00:45:46,944 --> 00:45:47,610 PROFESSOR: Yeah. 850 00:45:47,610 --> 00:45:49,020 There are probably also no people. 851 00:45:49,020 --> 00:45:50,020 AUDIENCE: But there are. 852 00:45:50,020 --> 00:45:51,699 The birth rate is equal to the-- so you 853 00:45:51,699 --> 00:45:52,810 have a stable population. 854 00:45:52,810 --> 00:45:53,518 PROFESSOR: Right. 855 00:45:53,518 --> 00:45:55,190 So in the limit, as a goes to 0, it's 856 00:45:55,190 --> 00:45:57,280 true that the number of zombies goes to 0. 857 00:46:00,208 --> 00:46:02,605 AUDIENCE: So you just have to beat [INAUDIBLE]. 858 00:46:02,605 --> 00:46:03,230 PROFESSOR: Yes. 859 00:46:06,520 --> 00:46:08,270 This is a good situation where it's 860 00:46:08,270 --> 00:46:10,770 important to ask about whether your assumptions in the model 861 00:46:10,770 --> 00:46:14,811 are good before making public policy based on them. 862 00:46:14,811 --> 00:46:16,901 OK? 863 00:46:16,901 --> 00:46:17,400 Indeed. 864 00:46:22,410 --> 00:46:24,260 But can we go back just for a moment 865 00:46:24,260 --> 00:46:28,165 before we switch gears, to ask about this question of why 866 00:46:28,165 --> 00:46:36,270 it was that war might have favored these predator fish? 867 00:46:36,270 --> 00:46:38,730 And by favored, what he really measured, 868 00:46:38,730 --> 00:46:40,920 he measured the number of the frequency 869 00:46:40,920 --> 00:46:44,590 of these kind of predator fish at the markets. 870 00:46:44,590 --> 00:46:48,940 So it's somehow is maybe the ratio of those two, or so. 871 00:47:00,170 --> 00:47:01,010 Yeah. 872 00:47:01,010 --> 00:47:01,580 That's right. 873 00:47:04,350 --> 00:47:09,700 Yes, it could that the fisherman just-- yeah. 874 00:47:09,700 --> 00:47:12,070 So they could have fished in a different region. 875 00:47:12,070 --> 00:47:14,861 And so the explanation could be something-- 876 00:47:14,861 --> 00:47:16,860 AUDIENCE: Because it's not very well controlled. 877 00:47:16,860 --> 00:47:18,100 PROFESSOR: It's not a controlled experiment. 878 00:47:18,100 --> 00:47:20,590 That's why you need to have many different wars in order 879 00:47:20,590 --> 00:47:23,175 to average the importance of large data. 880 00:47:27,860 --> 00:47:31,182 AUDIENCE: That's going to be a great [INAUDIBLE]. 881 00:47:31,182 --> 00:47:31,890 PROFESSOR: Right. 882 00:47:31,890 --> 00:47:33,556 And then of course, in all this business 883 00:47:33,556 --> 00:47:36,500 you have to ask, well, which thing are you varying more 884 00:47:36,500 --> 00:47:39,010 or less, et cetera, et cetera. 885 00:47:39,010 --> 00:47:41,340 But let's say that to first order, 886 00:47:41,340 --> 00:47:42,715 the fishermen are going out there 887 00:47:42,715 --> 00:47:44,298 and they're just catching all the fish 888 00:47:44,298 --> 00:47:45,350 in an unbiased fashion. 889 00:47:45,350 --> 00:47:47,650 But the war is making it more difficult to fish. 890 00:47:47,650 --> 00:47:49,610 So not as many fishermen go out. 891 00:47:49,610 --> 00:47:53,025 And then what does that do to these parameters? 892 00:47:55,811 --> 00:47:56,310 Right. 893 00:47:56,310 --> 00:47:58,420 So it might increase a, because the prey are not 894 00:47:58,420 --> 00:47:59,970 getting caught as much. 895 00:47:59,970 --> 00:48:02,060 And it might do something also to the predator. 896 00:48:02,060 --> 00:48:02,560 What? 897 00:48:10,329 --> 00:48:12,120 So the predator fish are also being caught, 898 00:48:12,120 --> 00:48:14,120 cause they're showing up at the market. 899 00:48:14,120 --> 00:48:15,620 Right. 900 00:48:15,620 --> 00:48:21,415 So which other parameters does it change? 901 00:48:25,640 --> 00:48:26,625 So what's that? 902 00:48:26,625 --> 00:48:27,810 AUDIENCE: [INAUDIBLE] 903 00:48:27,810 --> 00:48:27,950 PROFESSOR: Right. 904 00:48:27,950 --> 00:48:28,890 So lets be clear. 905 00:48:28,890 --> 00:48:36,080 It's a war makes a go down. 906 00:48:36,080 --> 00:48:36,580 All right. 907 00:48:40,190 --> 00:48:42,480 And what other parameter might change? 908 00:48:45,240 --> 00:48:46,767 AUDIENCE: Fewer fishermen. 909 00:48:46,767 --> 00:48:47,350 PROFESSOR: Oh. 910 00:48:47,350 --> 00:48:47,850 I'm sorry. 911 00:48:47,850 --> 00:48:49,729 I'm sorry. 912 00:48:49,729 --> 00:48:50,270 You're right. 913 00:48:50,270 --> 00:48:50,930 You're right. 914 00:48:53,600 --> 00:48:55,660 I was starting to think about the next one. 915 00:48:55,660 --> 00:48:56,160 OK. 916 00:48:56,160 --> 00:48:57,270 So a goes up. 917 00:48:57,270 --> 00:49:00,110 And what also happens? 918 00:49:00,110 --> 00:49:00,960 And c goes down. 919 00:49:00,960 --> 00:49:05,720 And this is just because a is a growth rate. c is a death rate. 920 00:49:05,720 --> 00:49:09,750 So it's not that the fishermen are preferentially 921 00:49:09,750 --> 00:49:12,095 changing what they're doing between the prey 922 00:49:12,095 --> 00:49:15,079 fish and the predator fish, but it's 923 00:49:15,079 --> 00:49:16,870 just that these are defined different ways. 924 00:49:16,870 --> 00:49:17,720 Right? 925 00:49:17,720 --> 00:49:21,880 And in particular, what this means is that in times of war, 926 00:49:21,880 --> 00:49:24,490 Voltaire's argument to his future son-in-law 927 00:49:24,490 --> 00:49:29,130 was that what you expect is that the predator population should 928 00:49:29,130 --> 00:49:30,550 go up. 929 00:49:30,550 --> 00:49:32,700 And the prey population should go down. 930 00:49:32,700 --> 00:49:35,070 So the ratio of them should certainly 931 00:49:35,070 --> 00:49:37,859 shift in favor of the predator. 932 00:49:37,859 --> 00:49:39,400 Or maybe it's just that the fishermen 933 00:49:39,400 --> 00:49:42,280 didn't go into the deep regions of water or something, 934 00:49:42,280 --> 00:49:44,980 where predator fish like to hang out. 935 00:49:44,980 --> 00:49:48,370 But it's striking at least that a simple model like this 936 00:49:48,370 --> 00:49:51,380 can actually provide some insight. 937 00:50:00,510 --> 00:50:02,682 Any questions about where we are before we 938 00:50:02,682 --> 00:50:03,890 modify the model in some way? 939 00:50:15,270 --> 00:50:18,220 Now, the problem with this model is 940 00:50:18,220 --> 00:50:19,850 that it's making lots of assumptions 941 00:50:19,850 --> 00:50:23,430 that we very much think are not true. 942 00:50:23,430 --> 00:50:26,910 So we have a few of the assumptions up there. 943 00:50:30,672 --> 00:50:32,130 Which ones are you guys least happy 944 00:50:32,130 --> 00:50:35,310 about among those assumptions? 945 00:50:38,181 --> 00:50:38,680 Yeah. 946 00:50:38,680 --> 00:50:41,020 The second one, the prey grows exponentially, 947 00:50:41,020 --> 00:50:42,220 i.e., without bound. 948 00:50:44,790 --> 00:50:46,210 That's not physical. 949 00:50:46,210 --> 00:50:47,540 We should fix that. 950 00:50:47,540 --> 00:50:49,330 So let's go ahead and do that. 951 00:50:49,330 --> 00:50:49,910 All right. 952 00:50:49,910 --> 00:50:57,581 So real prey populations-- and all populations-- populations 953 00:50:57,581 --> 00:50:59,450 saturate. 954 00:50:59,450 --> 00:51:02,540 Don't go to infinity, right. 955 00:51:07,150 --> 00:51:10,540 We can just make a quick little fix to our model 956 00:51:10,540 --> 00:51:11,770 to capture this, right? 957 00:51:11,770 --> 00:51:12,780 X dot. 958 00:51:12,780 --> 00:51:16,160 So now it's going to be this ax. 959 00:51:16,160 --> 00:51:19,930 But what we can do is we can add a logistic term here. 960 00:51:23,130 --> 00:51:24,880 And we can leave everything else constant. 961 00:51:34,170 --> 00:51:36,590 So we just add this logistic row term. 962 00:51:36,590 --> 00:51:40,180 So now in the absence of a predator, what 963 00:51:40,180 --> 00:51:42,670 will happen is that the prey population saturates 964 00:51:42,670 --> 00:51:46,690 at some point, at some value k. 965 00:51:46,690 --> 00:51:48,430 Right? 966 00:51:48,430 --> 00:51:50,490 And k could be rather large, if you like. 967 00:51:53,610 --> 00:51:56,604 And I think this is the kind of situation 968 00:51:56,604 --> 00:51:58,020 where you say, oh well, that could 969 00:51:58,020 --> 00:52:00,060 be just a really modest change. 970 00:52:00,060 --> 00:52:02,110 Because it's primarily telling us about what 971 00:52:02,110 --> 00:52:04,310 happens kind of in the absence of the predator. 972 00:52:04,310 --> 00:52:10,400 But really, the dynamics around here could be unchanged. 973 00:52:10,400 --> 00:52:12,270 But what's striking is that because 974 00:52:12,270 --> 00:52:14,860 of this neutral stability of the model, 975 00:52:14,860 --> 00:52:19,110 any little change can lead to a qualitative change 976 00:52:19,110 --> 00:52:20,540 in the outcome at equilibrium. 977 00:52:20,540 --> 00:52:23,280 And in particular, adding a carrying capacity 978 00:52:23,280 --> 00:52:28,830 causes the sustained oscillations to disappear. 979 00:52:28,830 --> 00:52:30,365 Instead you get damped oscillations. 980 00:52:33,320 --> 00:52:38,090 So in this case, xy, you still have this fixed point. 981 00:52:38,090 --> 00:52:44,790 But what happens is that you get-- it orbits the look. 982 00:52:44,790 --> 00:52:46,700 Kind of like this. 983 00:52:46,700 --> 00:52:47,220 Yes. 984 00:52:47,220 --> 00:52:48,136 AUDIENCE: [INAUDIBLE]. 985 00:52:50,769 --> 00:52:51,560 PROFESSOR: Yeah OK. 986 00:52:51,560 --> 00:52:54,870 So that depends upon [INAUDIBLE]. 987 00:52:54,870 --> 00:52:58,590 And indeed, as k goes to infinity, 988 00:52:58,590 --> 00:53:02,616 the timescale of this decay goes to infinity as well. 989 00:53:02,616 --> 00:53:07,250 AUDIENCE: [INAUDIBLE] clean scaling. 990 00:53:07,250 --> 00:53:08,984 PROFESSOR: A clean scaling. 991 00:53:08,984 --> 00:53:10,567 AUDIENCE: But I mean just [INAUDIBLE]. 992 00:53:13,075 --> 00:53:18,820 PROFESSOR: I mean, I don't have nothing intelligent to say. 993 00:53:18,820 --> 00:53:23,970 But what I can say is that if you put in a k 994 00:53:23,970 --> 00:53:27,650 at values that are close to this fixed point, 995 00:53:27,650 --> 00:53:29,810 then the oscillations really disappear. 996 00:53:29,810 --> 00:53:32,240 It's really only as k goes way, way out. 997 00:53:40,890 --> 00:53:42,860 And I maybe won't do this calculation. 998 00:53:42,860 --> 00:53:45,151 But you can do it quickly. 999 00:53:45,151 --> 00:53:47,650 It's the kind of thing you can do in five minutes on an exam 1000 00:53:47,650 --> 00:53:48,332 or so, right? 1001 00:53:50,986 --> 00:53:52,610 But then the question you might ask me, 1002 00:53:52,610 --> 00:53:55,951 well what can you do in order to get these oscillations that we 1003 00:53:55,951 --> 00:53:56,450 really like? 1004 00:53:56,450 --> 00:53:59,410 These sustained limit cycle oscillations? 1005 00:53:59,410 --> 00:54:03,370 And there are two ways you can think about this. 1006 00:54:03,370 --> 00:54:05,610 One is by a kind of taking this sort of model 1007 00:54:05,610 --> 00:54:08,680 and adding features and playing with it and so forth. 1008 00:54:08,680 --> 00:54:12,060 Or you can ask a mathematician who kind of derives everything 1009 00:54:12,060 --> 00:54:14,200 in kind of totality. 1010 00:54:14,200 --> 00:54:16,500 And then you can see it. 1011 00:54:16,500 --> 00:54:19,650 I'll say that one thing you can do on a concrete basis 1012 00:54:19,650 --> 00:54:23,770 to change the outcome of this is you can add something that 1013 00:54:23,770 --> 00:54:29,570 is kind of like some saturation effect 1014 00:54:29,570 --> 00:54:34,350 at the level of the interaction between predator and prey. 1015 00:54:34,350 --> 00:54:40,430 So instead of scaling purely as x times y, 1016 00:54:40,430 --> 00:54:46,600 if instead there some sort of Michaelis-Menten type behavior 1017 00:54:46,600 --> 00:54:50,500 as a function of the prey, then this model 1018 00:54:50,500 --> 00:54:56,360 is actually converted to one that has a stable limit cycle. 1019 00:54:56,360 --> 00:54:59,190 And in this case, can you guys remember 1020 00:54:59,190 --> 00:55:01,892 how you show that something has a stable limit cycle? 1021 00:55:14,784 --> 00:55:15,284 Yes. 1022 00:55:15,284 --> 00:55:18,260 AUDIENCE: [INAUDIBLE] use the Poincare-- 1023 00:55:18,260 --> 00:55:20,090 PROFESSOR: That's right. 1024 00:55:20,090 --> 00:55:22,730 There's this Poincare-Bendixson theorem, 1025 00:55:22,730 --> 00:55:26,630 which tells us that if we can draw some box out here 1026 00:55:26,630 --> 00:55:30,025 where the trajectories are all kind of coming in, indeed, here 1027 00:55:30,025 --> 00:55:31,650 the trajectories are kind of coming in. 1028 00:55:36,020 --> 00:55:38,230 Now in this case the trajectories 1029 00:55:38,230 --> 00:55:39,830 have to come in everywhere. 1030 00:55:39,830 --> 00:55:41,930 But this is typically true in any 1031 00:55:41,930 --> 00:55:44,380 of these reasonable systems. 1032 00:55:44,380 --> 00:55:48,120 Because they should, if there's just not that much food, 1033 00:55:48,120 --> 00:55:49,850 they should be coming in. 1034 00:55:49,850 --> 00:55:52,962 Now in this case, there's a single fixed point. 1035 00:55:52,962 --> 00:55:55,170 So then the question of whether there's a limit cycle 1036 00:55:55,170 --> 00:55:57,450 is equivalent to the question of the stability 1037 00:55:57,450 --> 00:55:59,140 of this interior fixed point. 1038 00:55:59,140 --> 00:56:02,080 It's a stable fixed point, then all of these trajectories 1039 00:56:02,080 --> 00:56:03,250 will spiral in. 1040 00:56:06,420 --> 00:56:08,310 However, if it's an unstable fixed point, 1041 00:56:08,310 --> 00:56:10,685 then you have to have a limit cycle oscillation somewhere 1042 00:56:10,685 --> 00:56:11,990 in between. 1043 00:56:11,990 --> 00:56:13,148 Yes. 1044 00:56:13,148 --> 00:56:16,016 AUDIENCE: It could be that it's a stable midpoint. 1045 00:56:16,016 --> 00:56:19,840 And then you have two limit cycles. 1046 00:56:19,840 --> 00:56:22,202 Could that be? 1047 00:56:22,202 --> 00:56:22,910 PROFESSOR: Right. 1048 00:56:22,910 --> 00:56:26,370 So the question is whether there can be a stable limit 1049 00:56:26,370 --> 00:56:30,130 cycle and two limit cycles. 1050 00:56:30,130 --> 00:56:34,550 I think that that requires more fixed points. 1051 00:56:34,550 --> 00:56:40,500 Because let's say you had a limit cycle here. 1052 00:56:40,500 --> 00:56:42,750 That this is saying that the trajectories come in. 1053 00:56:42,750 --> 00:56:45,680 But it's also saying the trajectories are coming out. 1054 00:56:45,680 --> 00:56:48,620 I think that you have to have a fixed point inside here 1055 00:56:48,620 --> 00:56:53,120 in order to have these trajectories going-- 1056 00:56:53,120 --> 00:56:55,362 AUDIENCE: So what I'm saying [INAUDIBLE] 1057 00:56:55,362 --> 00:56:58,743 two circles, and way outside, all the trajectory 1058 00:56:58,743 --> 00:57:00,250 is [INAUDIBLE]. 1059 00:57:00,250 --> 00:57:02,124 But between the two circles, the trajectories 1060 00:57:02,124 --> 00:57:04,539 go from the outer circle to the inner circle. 1061 00:57:04,539 --> 00:57:08,370 And then from the inner circle in [INAUDIBLE]. 1062 00:57:08,370 --> 00:57:10,750 PROFESSOR: Are the circles encircling each other? 1063 00:57:10,750 --> 00:57:11,697 AUDIENCE: Yeah. 1064 00:57:11,697 --> 00:57:12,280 PROFESSOR: OK. 1065 00:57:12,280 --> 00:57:13,380 So here's one circle. 1066 00:57:13,380 --> 00:57:15,670 Here's another circle. 1067 00:57:15,670 --> 00:57:19,950 And the trajectories are coming into this one 1068 00:57:19,950 --> 00:57:21,245 and what are they doing? 1069 00:57:21,245 --> 00:57:23,920 AUDIENCE: So each of these limit cycles are only semi-stable. 1070 00:57:23,920 --> 00:57:24,744 There are some-- 1071 00:57:24,744 --> 00:57:25,410 PROFESSOR: Yeah. 1072 00:57:25,410 --> 00:57:25,909 All right. 1073 00:57:25,909 --> 00:57:26,640 All right. 1074 00:57:26,640 --> 00:57:28,390 So you're talking about these limit cycles 1075 00:57:28,390 --> 00:57:31,260 where the trajectories come to it from one side, 1076 00:57:31,260 --> 00:57:33,000 but then leave from the other side. 1077 00:57:36,650 --> 00:57:38,750 I'd have to think about this more. 1078 00:57:38,750 --> 00:57:40,300 Everything that I'm talking about 1079 00:57:40,300 --> 00:57:42,897 are the limit cycles that are stable from both sides. 1080 00:57:42,897 --> 00:57:45,230 And I think that in the presence of any amount of noise, 1081 00:57:45,230 --> 00:57:47,790 those are the only ones that we care about. 1082 00:57:47,790 --> 00:57:51,791 Because you can write these models where you have orbits 1083 00:57:51,791 --> 00:57:53,790 that are stable from one side but not the other. 1084 00:57:53,790 --> 00:57:58,169 But then they don't matter in any real system 1085 00:57:58,169 --> 00:57:59,210 that is subject to noise. 1086 00:57:59,210 --> 00:58:01,590 Because it's not a stable orbit in that case, right? 1087 00:58:01,590 --> 00:58:04,412 AUDIENCE: Then we can make the outer circle stable 1088 00:58:04,412 --> 00:58:06,570 from both sides, and the inner circle [INAUDIBLE] 1089 00:58:06,570 --> 00:58:11,080 from both sides, and we could get it [INAUDIBLE]. 1090 00:58:11,080 --> 00:58:13,010 PROFESSOR: OK So it could go like this. 1091 00:58:13,010 --> 00:58:14,885 So you're saying that this outer orbit is now 1092 00:58:14,885 --> 00:58:16,130 stable from both sides. 1093 00:58:16,130 --> 00:58:19,620 And then this one is unstable from here. 1094 00:58:19,620 --> 00:58:23,090 And oh, OK. 1095 00:58:23,090 --> 00:58:25,940 Now you're saying this thing is unstable from-- so now 1096 00:58:25,940 --> 00:58:28,528 it's stable. 1097 00:58:28,528 --> 00:58:29,986 AUDIENCE: So we can have-- 1098 00:58:34,965 --> 00:58:35,590 PROFESSOR: Yes. 1099 00:58:35,590 --> 00:58:35,790 OK. 1100 00:58:35,790 --> 00:58:37,510 So I can certainly draw the trajectories, 1101 00:58:37,510 --> 00:58:38,470 as you pointed out. 1102 00:58:38,470 --> 00:58:39,220 And it works. 1103 00:58:42,781 --> 00:58:43,280 Yeah. 1104 00:58:43,280 --> 00:58:44,696 I don't know if this is a loophole 1105 00:58:44,696 --> 00:58:47,500 in the wording of the theorem, or if this is just not 1106 00:58:47,500 --> 00:58:48,390 possible. 1107 00:58:48,390 --> 00:58:53,850 AUDIENCE: [INAUDIBLE] There's like one way in which it 1108 00:58:53,850 --> 00:58:54,350 applies. 1109 00:58:54,350 --> 00:58:56,400 And then the other way you kind of usually 1110 00:58:56,400 --> 00:58:58,080 think that it applies, but then you're 1111 00:58:58,080 --> 00:59:00,038 wrong because there are these counter examples. 1112 00:59:00,038 --> 00:59:00,920 PROFESSOR: I see. 1113 00:59:00,920 --> 00:59:03,230 Well, how about this. 1114 00:59:03,230 --> 00:59:04,180 I will look it up. 1115 00:59:04,180 --> 00:59:08,520 And then I will tell you what I think the answer is. 1116 00:59:08,520 --> 00:59:13,270 I'm certainly not going to be able to derive it on the spot. 1117 00:59:13,270 --> 00:59:15,970 But You're right, though. 1118 00:59:15,970 --> 00:59:19,042 I can draw the orbits and they do it. 1119 00:59:19,042 --> 00:59:20,250 I don't know what that means. 1120 00:59:24,612 --> 00:59:26,070 AUDIENCE: So I think the theorem is 1121 00:59:26,070 --> 00:59:28,171 if on this large circle on the outside everything 1122 00:59:28,171 --> 00:59:30,435 is coming in, then if your fixed point is unstable, 1123 00:59:30,435 --> 00:59:32,894 then you must have at least one stable [INAUDIBLE] outside. 1124 00:59:32,894 --> 00:59:34,935 Or you must have at least one that is [INAUDIBLE] 1125 00:59:34,935 --> 00:59:35,970 or something like that. 1126 00:59:35,970 --> 00:59:36,553 PROFESSOR: OK. 1127 00:59:36,553 --> 00:59:38,110 But you're saying that it-- 1128 00:59:38,110 --> 00:59:41,400 AUDIENCE: But if you flip it so the fixed point is stable, 1129 00:59:41,400 --> 00:59:44,182 then you could have-- 1130 00:59:44,182 --> 00:59:44,890 PROFESSOR: I see. 1131 00:59:44,890 --> 00:59:45,980 That may be right. 1132 00:59:48,810 --> 00:59:51,460 I can't remember which direction. 1133 00:59:54,940 --> 00:59:58,370 But I'll look it up. 1134 00:59:58,370 --> 01:00:00,950 But if you are curious about the situations in which 1135 01:00:00,950 --> 01:00:04,310 these predator/prey functions will have limit cycles, 1136 01:00:04,310 --> 01:00:07,730 you can look up Kolmogorov's conditions. 1137 01:00:07,730 --> 01:00:11,440 So he basically has these four conditions 1138 01:00:11,440 --> 01:00:13,430 in which you will get a stable limit 1139 01:00:13,430 --> 01:00:16,140 cycle in the predator/prey populations. 1140 01:00:16,140 --> 01:00:18,640 And maybe I will. 1141 01:00:18,640 --> 01:00:20,450 And they're surprisingly simple. 1142 01:00:32,039 --> 01:00:33,080 They were in the reading. 1143 01:00:33,080 --> 01:00:34,210 So you can look at them. 1144 01:00:34,210 --> 01:00:37,790 But they're basically just these questions of the derivatives. 1145 01:00:37,790 --> 01:00:40,730 And they're derivatives of the per capita growth rates. 1146 01:00:40,730 --> 01:00:43,010 So you take those functions, you divide by x. 1147 01:00:43,010 --> 01:00:44,250 So it's x dot over x. 1148 01:00:44,250 --> 01:00:47,230 And then it's some f, y dot over y, sub g. 1149 01:00:47,230 --> 01:00:51,740 And the derivatives of those functions withe 1150 01:00:51,740 --> 01:00:53,570 respect to x and y have to be something, 1151 01:00:53,570 --> 01:00:55,050 at least in some limits. 1152 01:00:55,050 --> 01:00:58,990 And then you get limit cycles. 1153 01:00:58,990 --> 01:01:01,150 But I want to talk about the experiments. 1154 01:01:01,150 --> 01:01:03,450 Because I think that they were quite pretty. 1155 01:01:07,160 --> 01:01:09,740 So this is a paper by Yoshida, et al. 1156 01:01:16,290 --> 01:01:27,340 This is Nature 2003. 1157 01:01:27,340 --> 01:01:30,380 And the title was "Rapid Evolution Drives Ecological 1158 01:01:30,380 --> 01:01:33,031 Dynamics in a Predator/ Prey System." 1159 01:01:33,031 --> 01:01:34,780 Could somebody kind of say what they think 1160 01:01:34,780 --> 01:01:37,020 is the big point of this paper? 1161 01:01:50,936 --> 01:01:55,560 Do you guys like the paper, dislike the paper? 1162 01:01:55,560 --> 01:01:56,210 Too long? 1163 01:01:56,210 --> 01:01:57,330 Was it three pages? 1164 01:02:00,304 --> 01:02:02,470 Although it's really only-- I mean, one of the pages 1165 01:02:02,470 --> 01:02:03,870 is figures. 1166 01:02:03,870 --> 01:02:06,370 And one of the pages is essentially methods. 1167 01:02:06,370 --> 01:02:09,000 So it's really basically a one-page paper, 1168 01:02:09,000 --> 01:02:11,550 so it's not such heavy lifting, maybe. 1169 01:02:19,340 --> 01:02:19,840 All right. 1170 01:02:19,840 --> 01:02:21,970 Maybe I'll be concrete. 1171 01:02:21,970 --> 01:02:25,230 What features of their predator/prey oscillations 1172 01:02:25,230 --> 01:02:27,876 were different from a normal predator/prey oscillation? 1173 01:02:27,876 --> 01:02:30,000 What were the features they were trying to explain? 1174 01:02:37,175 --> 01:02:40,607 AUDIENCE: There's a longer phase life between the maximum. 1175 01:02:40,607 --> 01:02:42,940 PROFESSOR: And this is an interesting history, actually, 1176 01:02:42,940 --> 01:02:47,970 that these are authors-- they took what I think are basically 1177 01:02:47,970 --> 01:02:51,200 like samples from the Great Lakes, or something like that. 1178 01:02:51,200 --> 01:02:53,210 So they took a predator and prey. 1179 01:02:53,210 --> 01:02:59,070 So it's a rotifer algal system. 1180 01:02:59,070 --> 01:03:02,920 So the rotifer eats algae. 1181 01:03:05,840 --> 01:03:09,937 They had previously published a paper just a few years before, 1182 01:03:09,937 --> 01:03:11,770 where it was called something like "Crossing 1183 01:03:11,770 --> 01:03:14,640 the Hopf Bifurcation in a Predator/Prey System." 1184 01:03:14,640 --> 01:03:17,380 And so what they did is they had a chemostat where 1185 01:03:17,380 --> 01:03:20,330 there was some constant rate of dilution in their chemostat. 1186 01:03:20,330 --> 01:03:24,100 And then they measured the dynamics between the predator 1187 01:03:24,100 --> 01:03:25,780 and the prey. 1188 01:03:25,780 --> 01:03:27,980 And they showed that depending upon, 1189 01:03:27,980 --> 01:03:30,670 for example, the nutrients or this dilution 1190 01:03:30,670 --> 01:03:33,240 rate in their chemostat, they could go from a situation 1191 01:03:33,240 --> 01:03:38,066 where you have a stable coexistence-- 1192 01:03:38,066 --> 01:03:40,190 so a stable fixed point between the predator/prey-- 1193 01:03:40,190 --> 01:03:42,510 but at increasing dilution rate, they started 1194 01:03:42,510 --> 01:03:45,990 getting these oscillations. 1195 01:03:45,990 --> 01:03:49,070 So they went from stable to oscillations. 1196 01:03:49,070 --> 01:03:50,880 And then at higher dilution rates, 1197 01:03:50,880 --> 01:03:54,330 they got collapse of this predator/prey system. 1198 01:03:54,330 --> 01:03:56,590 So this is crossing the so-called Hopf bifurcation, 1199 01:03:56,590 --> 01:03:59,350 where you go from stable to unstable. 1200 01:03:59,350 --> 01:04:03,440 So they could actually see that they could use a simple model 1201 01:04:03,440 --> 01:04:04,910 to try to get some insight into why 1202 01:04:04,910 --> 01:04:06,737 it is that a predator/prey system might oscillate 1203 01:04:06,737 --> 01:04:07,950 or it might not oscillate. 1204 01:04:10,880 --> 01:04:12,040 And so that was great. 1205 01:04:12,040 --> 01:04:13,790 But there were some features in their data 1206 01:04:13,790 --> 01:04:16,220 that they didn't understand. 1207 01:04:16,220 --> 01:04:21,090 So the features were that the oscillations 1208 01:04:21,090 --> 01:04:22,280 had a really long period. 1209 01:04:25,030 --> 01:04:26,100 Long period oscillations. 1210 01:04:35,360 --> 01:04:37,820 And in all this data, it's a little bit hard 1211 01:04:37,820 --> 01:04:42,550 to get the exact phase lag, but in many of the cases 1212 01:04:42,550 --> 01:04:45,830 they saw that the predator and prey were not 1213 01:04:45,830 --> 01:04:49,380 at this 90 degrees out of phase like you would expect. 1214 01:04:49,380 --> 01:04:51,930 And in particular, for the predator/prey, 1215 01:04:51,930 --> 01:04:56,380 given that you have x and y, and it's going around some circle, 1216 01:04:56,380 --> 01:05:01,840 the idea is that first the prey population peaks. 1217 01:05:01,840 --> 01:05:07,080 And then kind of 90 degrees later phase, 1218 01:05:07,080 --> 01:05:09,960 you get a peak of the predator. 1219 01:05:09,960 --> 01:05:15,190 And you can imagine that in any predator/prey model 1220 01:05:15,190 --> 01:05:17,265 that you write down, not just Lotka-Volterra, 1221 01:05:17,265 --> 01:05:19,140 but if you write down a differential equation 1222 01:05:19,140 --> 01:05:23,980 with some x and some y, it's hard to get something 1223 01:05:23,980 --> 01:05:25,800 where it looks much different. 1224 01:05:25,800 --> 01:05:28,740 Of course, you can imagine maybe funny orbits or something 1225 01:05:28,740 --> 01:05:29,920 like that. 1226 01:05:29,920 --> 01:05:36,040 But generically you really expect this 90 degree phase 1227 01:05:36,040 --> 01:05:38,320 lag of the predator relative to the prey. 1228 01:05:41,137 --> 01:05:42,720 Whereas in the experiments, they often 1229 01:05:42,720 --> 01:05:57,204 saw 180 degree lag, lack of predator relative to the prey. 1230 01:05:57,204 --> 01:05:57,704 Yes. 1231 01:05:57,704 --> 01:06:00,184 AUDIENCE: So when you say [INAUDIBLE] theory 1232 01:06:00,184 --> 01:06:02,664 of oscillation, what does that mean? 1233 01:06:02,664 --> 01:06:04,650 What's the time scale? 1234 01:06:04,650 --> 01:06:05,517 PROFESSOR: Yeah. 1235 01:06:05,517 --> 01:06:08,380 AUDIENCE: What would make this suprisingly [INAUDIBLE]? 1236 01:06:08,380 --> 01:06:09,251 PROFESSOR: Right. 1237 01:06:09,251 --> 01:06:09,750 Yeah. 1238 01:06:09,750 --> 01:06:09,930 OK. 1239 01:06:09,930 --> 01:06:11,805 So this is really in the context of-- I mean, 1240 01:06:11,805 --> 01:06:14,800 they wrote down a model where they did basically have 1241 01:06:14,800 --> 01:06:18,514 something like this, where they said there's some-- 1242 01:06:18,514 --> 01:06:21,340 AUDIENCE: [INAUDIBLE] 1243 01:06:21,340 --> 01:06:22,830 PROFESSOR: That's right. 1244 01:06:22,830 --> 01:06:25,390 And in particular the predator, I think, 1245 01:06:25,390 --> 01:06:27,260 was not really dying on it's own. 1246 01:06:27,260 --> 01:06:29,300 Except for the dilution. 1247 01:06:29,300 --> 01:06:32,920 So it's really the dilution rate from the chemostat gave them c. 1248 01:06:32,920 --> 01:06:35,570 And they could see that indeed, the rotifer wasn't dying much. 1249 01:06:35,570 --> 01:06:37,150 And then a is telling you about what 1250 01:06:37,150 --> 01:06:39,834 happens if you don't have the rotifer, 1251 01:06:39,834 --> 01:06:41,000 you don't have the predator. 1252 01:06:41,000 --> 01:06:42,680 And then at that dilution rate, you 1253 01:06:42,680 --> 01:06:45,050 ask what the division rate is. 1254 01:06:45,050 --> 01:06:47,530 And that's essentially the growth rate 1255 01:06:47,530 --> 01:06:50,360 of that prey population minus the dilution 1256 01:06:50,360 --> 01:06:51,560 rate of the chemostat. 1257 01:06:51,560 --> 01:06:54,880 I'd say that they had these parameters pretty well. 1258 01:07:01,620 --> 01:07:02,120 Yes. 1259 01:07:02,120 --> 01:07:07,740 And what they saw experimentally was that it took much longer. 1260 01:07:07,740 --> 01:07:12,264 I mean, the period was 5 times, 10 times longer 1261 01:07:12,264 --> 01:07:13,680 than what they would have expected 1262 01:07:13,680 --> 01:07:17,810 based on the measured parameters for the predator and the prey. 1263 01:07:24,980 --> 01:07:27,140 And of course, that was annoying for them, 1264 01:07:27,140 --> 01:07:29,400 because they're actually doing the experiments. 1265 01:07:29,400 --> 01:07:31,483 So they had to do these chemostat experiments that 1266 01:07:31,483 --> 01:07:33,660 lasted six months or something like that, in order 1267 01:07:33,660 --> 01:07:35,770 to see a couple oscillations. 1268 01:07:35,770 --> 01:07:40,120 And so this longer period wasn't just a mathematical-- 1269 01:07:40,120 --> 01:07:42,270 I'm sure that they were disappointed to see 1270 01:07:42,270 --> 01:07:44,734 these really long time-scale oscillations, because it was 1271 01:07:44,734 --> 01:07:46,650 something that made it very difficult for them 1272 01:07:46,650 --> 01:07:47,608 to do the measurements. 1273 01:07:51,940 --> 01:07:54,670 But this is what they saw experimentally. 1274 01:07:54,670 --> 01:07:57,970 And they didn't know what was causing it. 1275 01:07:57,970 --> 01:08:02,600 But they later then did more modeling 1276 01:08:02,600 --> 01:08:05,880 where they asked, which of the assumptions 1277 01:08:05,880 --> 01:08:09,200 in our original model might not be true. 1278 01:08:09,200 --> 01:08:10,685 And the nature of these things is 1279 01:08:10,685 --> 01:08:12,060 that there are an infinite number 1280 01:08:12,060 --> 01:08:16,380 of things are true that are not incorporated in the model. 1281 01:08:16,380 --> 01:08:17,939 And this makes it quite challenging 1282 01:08:17,939 --> 01:08:19,569 to isolate what the effects might be. 1283 01:08:19,569 --> 01:08:22,960 But at the very least what they can do is they can go in. 1284 01:08:22,960 --> 01:08:24,944 And they have some sense of their system. 1285 01:08:24,944 --> 01:08:26,319 And they can make guesses of what 1286 01:08:26,319 --> 01:08:27,960 might have been the primary things missing. 1287 01:08:27,960 --> 01:08:29,290 And then they kind of went through the models 1288 01:08:29,290 --> 01:08:30,749 and they asked, well, if we kind of 1289 01:08:30,749 --> 01:08:32,498 change this assumption or that assumption, 1290 01:08:32,498 --> 01:08:34,439 what does it do in terms of the oscillations. 1291 01:08:34,439 --> 01:08:37,729 And then what they found in a modeling paper-- 1292 01:08:37,729 --> 01:08:41,779 so this is their previous experimental paper before this. 1293 01:08:41,779 --> 01:08:46,020 What they found in their models was 1294 01:08:46,020 --> 01:08:50,259 that prey evolution was kind of the one thing 1295 01:08:50,259 --> 01:08:52,100 that if they allowed prey evolution, i.e. 1296 01:08:52,100 --> 01:08:54,450 If they allowed different types of prey 1297 01:08:54,450 --> 01:08:56,029 in their prey population that might 1298 01:08:56,029 --> 01:08:58,840 be oscillating independently, then they 1299 01:08:58,840 --> 01:09:00,670 could get both of these effects. 1300 01:09:00,670 --> 01:09:02,240 So from the models they concluded 1301 01:09:02,240 --> 01:09:07,420 that prey evolution could give this, 1302 01:09:07,420 --> 01:09:09,950 could maybe explain those two things. 1303 01:09:14,613 --> 01:09:16,279 These are the two experimental features. 1304 01:09:19,819 --> 01:09:21,064 Yeah. 1305 01:09:21,064 --> 01:09:22,540 AUDIENCE: It might be that you have 1306 01:09:22,540 --> 01:09:24,508 [INAUDIBLE] symantic point. 1307 01:09:24,508 --> 01:09:29,889 If you have two types of species, pre-species-- 1308 01:09:29,889 --> 01:09:34,116 [INAUDIBLE] they have these characteristics 1309 01:09:34,116 --> 01:09:36,044 and they're not evolving. 1310 01:09:36,044 --> 01:09:37,490 They're just fixed. 1311 01:09:37,490 --> 01:09:38,356 PROFESSOR: Yes. 1312 01:09:38,356 --> 01:09:39,939 AUDIENCE: You would still be able to-- 1313 01:09:39,939 --> 01:09:40,479 PROFESSOR: Exactly. 1314 01:09:40,479 --> 01:09:40,979 Right. 1315 01:09:40,979 --> 01:09:41,826 This is a major-- 1316 01:09:41,826 --> 01:09:42,700 AUDIENCE: [INAUDIBLE] 1317 01:09:42,700 --> 01:09:43,160 PROFESSOR: Yes. 1318 01:09:43,160 --> 01:09:45,618 You can argue about what you want to call evolution and not 1319 01:09:45,618 --> 01:09:47,359 revolution. 1320 01:09:47,359 --> 01:09:51,350 Over these time scales what they are not looking at 1321 01:09:51,350 --> 01:09:55,070 is the de novo kind of emergence of new mutants spreading 1322 01:09:55,070 --> 01:09:55,990 in the population. 1323 01:09:55,990 --> 01:10:00,410 What they're looking at is variations in say, 1324 01:10:00,410 --> 01:10:03,703 different types of prey. 1325 01:10:03,703 --> 01:10:08,020 AUDIENCE: So is that what we would call evolution? 1326 01:10:08,020 --> 01:10:09,450 PROFESSOR: Well, OK. 1327 01:10:09,450 --> 01:10:10,700 This is a major question. 1328 01:10:10,700 --> 01:10:13,926 I think different people call evolution different things. 1329 01:10:13,926 --> 01:10:15,550 And in particular there's a distinction 1330 01:10:15,550 --> 01:10:22,100 between the ecologists and the evolutionary biologists, 1331 01:10:22,100 --> 01:10:24,250 or evolutionary microbiologists, or whatnot. 1332 01:10:24,250 --> 01:10:29,390 And this is something that I certainly encounter. 1333 01:10:29,390 --> 01:10:32,310 For the experimental microbial evolution guys, 1334 01:10:32,310 --> 01:10:35,224 I'd say that for them, if you use the word "evolution," what 1335 01:10:35,224 --> 01:10:36,890 they really want to see, or what they're 1336 01:10:36,890 --> 01:10:39,930 expecting to see is kind of new mutants arising 1337 01:10:39,930 --> 01:10:42,479 in the population, spreading, fixing, 1338 01:10:42,479 --> 01:10:44,520 and that changes the character of the population. 1339 01:10:44,520 --> 01:10:47,110 Whereas from that standpoint of ecologists-- 1340 01:10:47,110 --> 01:10:50,880 and indeed, if you look up the definition of evolution, 1341 01:10:50,880 --> 01:10:53,660 it's some change in allele frequency over time. 1342 01:10:53,660 --> 01:10:56,200 And this could certainly be a change allele frequency 1343 01:10:56,200 --> 01:10:59,010 over time in the sense that you could imagine this arising just 1344 01:10:59,010 --> 01:11:03,630 from, for example, there could be a point mutation 1345 01:11:03,630 --> 01:11:06,470 in some gene that leads it to do something or another. 1346 01:11:06,470 --> 01:11:08,070 And in this paper they didn't identify 1347 01:11:08,070 --> 01:11:09,070 exactly what was going on. 1348 01:11:09,070 --> 01:11:11,236 I'll tell you a little bit about a later paper where 1349 01:11:11,236 --> 01:11:12,540 they get a better sense of it. 1350 01:11:12,540 --> 01:11:14,540 But at least you could imagine it being the case 1351 01:11:14,540 --> 01:11:16,700 that there's just a point mutation, some gene that 1352 01:11:16,700 --> 01:11:19,750 has some fitness defect in the absence of the predator, 1353 01:11:19,750 --> 01:11:22,541 but then allows the algae to avoid the predator in some way. 1354 01:11:22,541 --> 01:11:25,490 AUDIENCE: [INAUDIBLE] 1355 01:11:25,490 --> 01:11:26,629 PROFESSOR: That's right. 1356 01:11:26,629 --> 01:11:27,170 That's right. 1357 01:11:27,170 --> 01:11:29,080 As far as the model goes, it could 1358 01:11:29,080 --> 01:11:31,800 be that the prey are really-- one species was just 1359 01:11:31,800 --> 01:11:32,550 a point mutation. 1360 01:11:32,550 --> 01:11:33,550 Then you'd say, oh yeah. 1361 01:11:33,550 --> 01:11:35,420 That's kind of evolution. 1362 01:11:35,420 --> 01:11:40,230 Or it could be two different prey populations and so forth. 1363 01:11:40,230 --> 01:11:40,730 Right? 1364 01:11:40,730 --> 01:11:44,330 And depending upon what information you have access to, 1365 01:11:44,330 --> 01:11:46,899 you would either be aware of these things or not. 1366 01:11:46,899 --> 01:11:48,940 In this case, they just look under the microscope 1367 01:11:48,940 --> 01:11:51,790 and say, oh yeah, this looks like an algae, or whatnot. 1368 01:11:51,790 --> 01:11:54,040 And maybe they know that it's the same species. 1369 01:11:54,040 --> 01:11:58,570 But they could be different in lots of different ways. 1370 01:11:58,570 --> 01:12:01,830 And I think this discussion is highlighting 1371 01:12:01,830 --> 01:12:07,290 that a lot of the same effects that you see in ecology also 1372 01:12:07,290 --> 01:12:09,420 you see in evolution and vice versa. 1373 01:12:09,420 --> 01:12:15,019 And depending upon your focus, you 1374 01:12:15,019 --> 01:12:17,060 might want to put it more in the bin of evolution 1375 01:12:17,060 --> 01:12:18,850 or in terms of population dynamics. 1376 01:12:18,850 --> 01:12:23,890 I think that's a matter of taste at some point, actually. 1377 01:12:23,890 --> 01:12:24,390 Yeah. 1378 01:12:24,390 --> 01:12:25,764 Because they say rapid evolution. 1379 01:12:25,764 --> 01:12:27,001 And I think-- 1380 01:12:29,800 --> 01:12:31,302 AUDIENCE: [INAUDIBLE]. 1381 01:12:31,302 --> 01:12:32,260 PROFESSOR: Yeah, right. 1382 01:12:32,260 --> 01:12:35,352 So I think that different people will 1383 01:12:35,352 --> 01:12:36,560 have different takes on this. 1384 01:12:36,560 --> 01:12:38,018 I think, once again, this is really 1385 01:12:38,018 --> 01:12:39,980 very much from the standpoint of an ecologist, 1386 01:12:39,980 --> 01:12:41,040 this is rapid evolution. 1387 01:12:41,040 --> 01:12:45,674 Because every cycle, the allele frequency is changing. 1388 01:12:45,674 --> 01:12:47,340 And that's what's maybe leading to this. 1389 01:12:47,340 --> 01:12:49,670 But the thing is, it's very easy for you 1390 01:12:49,670 --> 01:12:51,720 to read certainly the title, and come away 1391 01:12:51,720 --> 01:12:54,270 thinking it's something different from what it was. 1392 01:12:56,780 --> 01:12:57,403 I agree. 1393 01:13:00,930 --> 01:13:04,150 But one of the things I like about this whole line 1394 01:13:04,150 --> 01:13:06,720 of research that took place over the course of say, 15 years, 1395 01:13:06,720 --> 01:13:12,060 was that they did these nice measurements. 1396 01:13:12,060 --> 01:13:15,050 They're guided by models. 1397 01:13:15,050 --> 01:13:17,430 They saw some things that they were expecting, 1398 01:13:17,430 --> 01:13:18,960 some things they couldn't explain. 1399 01:13:18,960 --> 01:13:20,869 And then they went and did more modeling. 1400 01:13:20,869 --> 01:13:22,160 And then they could explain it. 1401 01:13:22,160 --> 01:13:26,162 And that guided this experiment. 1402 01:13:26,162 --> 01:13:27,870 Because before what they did is they just 1403 01:13:27,870 --> 01:13:30,954 took kind of the sludge, or whatever, from the lake. 1404 01:13:30,954 --> 01:13:32,370 So there were many different types 1405 01:13:32,370 --> 01:13:34,360 of algae and of the rotifer, for that matter. 1406 01:13:34,360 --> 01:13:36,610 And then they looked at the predator/prey oscillations 1407 01:13:36,610 --> 01:13:37,480 between those. 1408 01:13:37,480 --> 01:13:39,090 And then in this paper what they did 1409 01:13:39,090 --> 01:13:41,470 is they took individual prey populations that 1410 01:13:41,470 --> 01:13:42,990 came from a single prey. 1411 01:13:42,990 --> 01:13:45,619 So they were kind of isogenic. 1412 01:13:45,619 --> 01:13:48,160 Of course eventually they would evolve, and blah, blah, blah. 1413 01:13:48,160 --> 01:13:51,190 But in this case they took individual isolates 1414 01:13:51,190 --> 01:13:52,090 from the prey. 1415 01:13:52,090 --> 01:13:56,120 And then what they saw is that these two features went away. 1416 01:13:56,120 --> 01:13:58,540 The oscillation period was more what they were expecting. 1417 01:13:58,540 --> 01:14:01,740 And they got the 90 degree phase lag. 1418 01:14:01,740 --> 01:14:04,160 So it's was really a case of these models 1419 01:14:04,160 --> 01:14:13,720 led to the experiments where they took kind of clonal prey. 1420 01:14:13,720 --> 01:14:16,160 And then they recovered their classical predictions. 1421 01:14:27,440 --> 01:14:31,172 And indeed, they've recently done some other measurements 1422 01:14:31,172 --> 01:14:32,380 that I think were quite nice. 1423 01:14:32,380 --> 01:14:35,760 They had an ecology letters paper just a couple years ago 1424 01:14:35,760 --> 01:14:42,450 where what they did is they took two different algae that 1425 01:14:42,450 --> 01:14:47,310 had different types embodying these trade-offs that we 1426 01:14:47,310 --> 01:14:48,530 were talking about. 1427 01:14:48,530 --> 01:14:53,970 In particular, one of the algae can divide rapidly. 1428 01:14:53,970 --> 01:14:55,840 But it kind of exists as singles. 1429 01:14:55,840 --> 01:15:01,240 Another algae is kind of clumpy. 1430 01:15:01,240 --> 01:15:04,920 So this algal type was able to divide more rapidly 1431 01:15:04,920 --> 01:15:06,220 than this algal type. 1432 01:15:06,220 --> 01:15:12,070 But this type was harder to eat just because it was clumpy. 1433 01:15:12,070 --> 01:15:12,630 OK? 1434 01:15:12,630 --> 01:15:18,220 It's one of things that once you hear it, you say, oh yeah. 1435 01:15:18,220 --> 01:15:18,780 Sure. 1436 01:15:18,780 --> 01:15:24,510 But then in the abstract, when you read this 2003 paper, 1437 01:15:24,510 --> 01:15:27,720 you think, oh, I can't imagine what kind of behavior 1438 01:15:27,720 --> 01:15:31,190 could possibly help the algae avoid these rotifers. 1439 01:15:31,190 --> 01:15:34,710 But then in this ecology letters paper, what they did 1440 01:15:34,710 --> 01:15:38,070 is they actually tracked population densities 1441 01:15:38,070 --> 01:15:42,090 of this type, this type, and of the rotifer over time. 1442 01:15:42,090 --> 01:15:46,024 So then they could see all three sub-populations oscillating. 1443 01:15:46,024 --> 01:15:47,440 So then you can really kind of see 1444 01:15:47,440 --> 01:15:51,710 how this evolution, if you want, or you could just say it's 1445 01:15:51,710 --> 01:15:52,210 ecology. 1446 01:15:52,210 --> 01:15:54,620 Because it could be different species for all we care. 1447 01:15:56,970 --> 01:16:02,380 But in any case, it's certainly prey heterogeneity anyway you 1448 01:16:02,380 --> 01:16:02,880 look at it. 1449 01:16:02,880 --> 01:16:04,296 Whether it's evolution or ecology. 1450 01:16:04,296 --> 01:16:07,930 It's heterogeneity in the prey population that 1451 01:16:07,930 --> 01:16:11,194 leads to these very qualitatively 1452 01:16:11,194 --> 01:16:12,610 different population oscillations. 1453 01:16:17,730 --> 01:16:19,480 Any other questions on that before we-- 1454 01:16:23,050 --> 01:16:24,550 And so just for the last few minutes 1455 01:16:24,550 --> 01:16:26,620 I'll say something about this question 1456 01:16:26,620 --> 01:16:29,040 of noise-induced oscillations. 1457 01:16:29,040 --> 01:16:31,890 This is going to be something you're 1458 01:16:31,890 --> 01:16:35,630 going to be playing with over the next week. 1459 01:16:35,630 --> 01:16:39,940 So you'll be experts eventually. 1460 01:16:39,940 --> 01:16:42,300 And the discussion in the homework 1461 01:16:42,300 --> 01:16:44,680 is really guided by this paper by McKane and Newman. 1462 01:16:49,290 --> 01:16:50,830 All right. 1463 01:16:50,830 --> 01:16:51,570 So McKane. 1464 01:16:51,570 --> 01:16:53,200 Newman. 1465 01:16:53,200 --> 01:16:56,300 McKane is a professor at Manchester. 1466 01:16:56,300 --> 01:17:03,360 And what they showed is that if you take this model here, 1467 01:17:03,360 --> 01:17:06,560 did this model have limit cycle oscillations, 1468 01:17:06,560 --> 01:17:08,520 sustained oscillations? 1469 01:17:08,520 --> 01:17:09,560 No. 1470 01:17:09,560 --> 01:17:12,160 So this model where we include the carrying 1471 01:17:12,160 --> 01:17:13,920 capacity of the prey, this does not 1472 01:17:13,920 --> 01:17:15,380 have sustained oscillations. 1473 01:17:15,380 --> 01:17:23,350 So you take a model like this that just has a stable spiral. 1474 01:17:23,350 --> 01:17:26,200 Now this is a model in the context of-- it's 1475 01:17:26,200 --> 01:17:28,270 a differential equation model. 1476 01:17:28,270 --> 01:17:33,080 Now the question is, how do you incorporate noise? 1477 01:17:33,080 --> 01:17:35,730 Just the fact that individuals give birth, 1478 01:17:35,730 --> 01:17:37,140 they die at random times. 1479 01:17:39,860 --> 01:17:42,260 The first order way that we often think to do this is we 1480 01:17:42,260 --> 01:17:46,270 just add some noise on to the differential equations. 1481 01:17:46,270 --> 01:17:50,010 So what we do is, you might say, oh well, you could add some a 1482 01:17:50,010 --> 01:17:54,390 to x, and some a to y. 1483 01:17:54,390 --> 01:17:57,970 Maybe this noise should be proportional to x or something. 1484 01:17:57,970 --> 01:18:00,000 So you could think about the strength of this 1485 01:18:00,000 --> 01:18:02,180 might be some pre-factor. 1486 01:18:02,180 --> 01:18:05,550 So you could just take kind of a [INAUDIBLE] approach 1487 01:18:05,550 --> 01:18:08,480 where you add noise onto the differential equations. 1488 01:18:08,480 --> 01:18:11,800 And then what you get is a noisy kind 1489 01:18:11,800 --> 01:18:16,155 of path to that fixed point. 1490 01:18:16,155 --> 01:18:21,940 The perhaps surprising thing is that instead of adding noise 1491 01:18:21,940 --> 01:18:24,280 to a differential equation, if instead you 1492 01:18:24,280 --> 01:18:26,670 start with the individual based approach, 1493 01:18:26,670 --> 01:18:28,300 and you take kind of a master equation 1494 01:18:28,300 --> 01:18:30,810 approach where you say they're individuals 1495 01:18:30,810 --> 01:18:32,420 and they're doing something. 1496 01:18:32,420 --> 01:18:35,740 Individual predators are eating individual prey, et cetera. 1497 01:18:35,740 --> 01:18:44,160 So if instead you take a master equation, 1498 01:18:44,160 --> 01:18:48,210 just like what we did for the chemical 1499 01:18:48,210 --> 01:18:50,406 equations modeling the cells and so forth 1500 01:18:50,406 --> 01:18:52,530 that we talked about at the beginning of the class, 1501 01:18:52,530 --> 01:18:55,780 if you formulate this predator/prey system 1502 01:18:55,780 --> 01:19:00,460 as an underlying set of kind of individual based interactions, 1503 01:19:00,460 --> 01:19:04,720 then what you actually find is that you get surprisingly large 1504 01:19:04,720 --> 01:19:08,250 sustained oscillations. 1505 01:19:08,250 --> 01:19:09,800 So you really end up with a situation 1506 01:19:09,800 --> 01:19:11,350 where this thing comes. 1507 01:19:11,350 --> 01:19:14,580 And it's noisy, of course. 1508 01:19:14,580 --> 01:19:17,450 But it kind of comes around here in some way 1509 01:19:17,450 --> 01:19:20,200 that looks like this. 1510 01:19:20,200 --> 01:19:24,380 And you'll see this in your simulations next week where-- 1511 01:19:24,380 --> 01:19:26,750 and I don't know if you can see this at all. 1512 01:19:26,750 --> 01:19:30,590 But this is some plots of predator and prey 1513 01:19:30,590 --> 01:19:33,420 at reasonably large numbers, where you can actually 1514 01:19:33,420 --> 01:19:35,520 have 1,000 predator or prey. 1515 01:19:35,520 --> 01:19:39,880 Somehow you can get some sort of resonant enhancement 1516 01:19:39,880 --> 01:19:41,190 of these oscillations. 1517 01:19:41,190 --> 01:19:42,660 And kind of what's going on is that 1518 01:19:42,660 --> 01:19:46,720 the demographic fluctuations of the demographic noise 1519 01:19:46,720 --> 01:19:49,720 excites the system at all frequencies. 1520 01:19:49,720 --> 01:19:52,240 But then there is a characteristic frequency 1521 01:19:52,240 --> 01:19:56,150 given basically by this frequency 1522 01:19:56,150 --> 01:19:58,180 of the inherent oscillation there 1523 01:19:58,180 --> 01:20:00,500 that is somehow amplified. 1524 01:20:00,500 --> 01:20:02,000 So you end up with a situation where 1525 01:20:02,000 --> 01:20:03,125 you get these oscillations. 1526 01:20:06,690 --> 01:20:09,020 And they're noisy and whatnot. 1527 01:20:09,020 --> 01:20:13,570 But it's really because it's this particular frequency that 1528 01:20:13,570 --> 01:20:15,280 was amplified. 1529 01:20:15,280 --> 01:20:18,010 And just because it takes a long time for those 1530 01:20:18,010 --> 01:20:19,340 oscillations to go away. 1531 01:20:19,340 --> 01:20:23,580 So you can imagine that the amplitude of the oscillation 1532 01:20:23,580 --> 01:20:26,020 falls off as kind of a root end scaling, 1533 01:20:26,020 --> 01:20:29,130 because it's kind of a demographic type noise. 1534 01:20:29,130 --> 01:20:31,715 So it is true that the relative amplitude 1535 01:20:31,715 --> 01:20:33,940 of these oscillations, it gets larger as you 1536 01:20:33,940 --> 01:20:35,890 to smaller population sizes. 1537 01:20:35,890 --> 01:20:40,850 But it still ends up being a surprisingly large effect. 1538 01:20:40,850 --> 01:20:43,910 And you'll see how this plays out in this model that's 1539 01:20:43,910 --> 01:20:45,680 basically guided by this paper. 1540 01:20:45,680 --> 01:20:48,030 And this was a PRL in 2005. 1541 01:20:48,030 --> 01:20:49,650 Incidentally, other people have since 1542 01:20:49,650 --> 01:20:52,010 studied how these sorts of ideas can 1543 01:20:52,010 --> 01:20:56,935 result in noise-induced pattern formation, as well. 1544 01:20:56,935 --> 01:20:59,399 All right, with that, I will let you guys go. 1545 01:20:59,399 --> 01:21:00,440 Have a good Thanksgiving. 1546 01:21:00,440 --> 01:21:02,810 I'll see you guys on Tuesday.